{"id":112,"date":"2019-08-20T17:02:39","date_gmt":"2019-08-20T21:02:39","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/fitting-exponential-models-to-data\/"},"modified":"2022-06-01T10:39:29","modified_gmt":"2022-06-01T14:39:29","slug":"fitting-exponential-models-to-data","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/fitting-exponential-models-to-data\/","title":{"raw":"Fitting Exponential Models to Data","rendered":"Fitting Exponential Models to Data"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Build an exponential model from data.<\/li>\n \t<li>Build a logarithmic model from data.<\/li>\n \t<li>Build a logistic model from data.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1677675\">In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called <em>regression analysis<\/em> to find a curve that models data collected from real-world observations. With <span class=\"no-emphasis\">regression analysis<\/span>, we don\u2019t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events.<\/p>\n<p id=\"fs-id1356390\">Do not be confused by the word <em>model<\/em>. In mathematics, we often use the terms <em>function<\/em>, <em>equation<\/em>, and <em>model<\/em> interchangeably, even though they each have their own formal definition. The term <em>model<\/em> is typically used to indicate that the equation or function approximates a real-world situation.<\/p>\n<p id=\"fs-id969059\">We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work we\u2019ve done so far, and then explore the ways regression is used to model real-world phenomena.<\/p>\n\n<div id=\"fs-id1527075\" class=\"bc-section section\">\n<h3>Building an Exponential Model from Data<\/h3>\n<p id=\"fs-id1637870\">As we\u2019ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that\u2019s not the whole story. It\u2019s the <em>way<\/em> data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let\u2019s review exponential growth and decay.<\/p>\n<p id=\"fs-id1365833\">Recall that exponential functions have the form[latex]\\,y=a{b}^{x}\\,[\/latex]or[latex]\\,y={A}_{0}{e}^{kx}.\\,[\/latex]When performing regression analysis, we use the form most commonly used on graphing utilities,[latex]\\,y=a{b}^{x}.\\,[\/latex]Take a moment to reflect on the characteristics we\u2019ve already learned about the exponential function[latex]\\,y=a{b}^{x}\\,[\/latex](assume[latex]\\,a&gt;0[\/latex]):<\/p>\n\n<ul id=\"fs-id1694862\">\n \t<li>[latex]b\\,[\/latex]must be greater than zero and not equal to one.<\/li>\n \t<li>The initial value of the model is[latex]\\,y=a.[\/latex]\n<ul id=\"fs-id1305009\">\n \t<li>If[latex]\\,b&gt;1,[\/latex] the function models exponential growth. As[latex]\\,x\\,[\/latex]increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound.<\/li>\n \t<li>If[latex]\\,0&lt;b&lt;1,[\/latex] the function models <span class=\"no-emphasis\">exponential decay<\/span>. As[latex]\\,x\\,[\/latex]increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the <em>x<\/em>-axis. In other words, the outputs never become equal to or less than zero.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p id=\"fs-id1373960\">As part of the results, your calculator will display a number known as the <em>correlation coefficient<\/em>, labeled by the variable[latex]\\,r,[\/latex] or[latex]\\,{r}^{2}.\\,[\/latex](You may have to change the calculator\u2019s settings for these to be shown.) The values are an indication of the \u201cgoodness of fit\u201d of the regression equation to the data. We more commonly use the value of[latex]\\,{r}^{2}\\,[\/latex]instead of[latex]\\,r,[\/latex] but the closer either value is to 1, the better the regression equation approximates the data.<\/p>\n\n<div id=\"fs-id1424609\" class=\"textbox key-takeaways\">\n<h3>Exponential Regression<\/h3>\n<p id=\"fs-id1532328\"><em>Exponential regression<\/em> is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command \u201cExpReg\u201d on a graphing utility to fit an exponential function to a set of data points. This returns an equation of the form,[latex]y=a{b}^{x}[\/latex]<\/p>\n<p id=\"fs-id1585350\">Note that:<\/p>\n\n<ul id=\"fs-id1534713\">\n \t<li>[latex]b\\,[\/latex]must be non-negative.<\/li>\n \t<li>when[latex]\\,b&gt;1,[\/latex] we have an exponential growth model.<\/li>\n \t<li>when[latex]\\,0&lt;b&lt;1,[\/latex] we have an exponential decay model.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1375157\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1588505\"><strong>Given a set of data, perform exponential regression using a graphing utility.<\/strong><\/p>\n\n<ol id=\"fs-id836076\" type=\"1\">\n \t<li>Use the STAT then EDIT menu to enter given data.\n<ol id=\"fs-id1293089\" type=\"a\">\n \t<li>Clear any existing data from the lists.<\/li>\n \t<li>List the input values in the L1 column.<\/li>\n \t<li>List the output values in the L2 column.<\/li>\n<\/ol>\n<\/li>\n \t<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\n<ol id=\"fs-id900017\" type=\"a\">\n \t<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\n \t<li>Verify the data follow an exponential pattern.<\/li>\n<\/ol>\n<\/li>\n \t<li>Find the equation that models the data.\n<ol id=\"fs-id1677252\" type=\"a\">\n \t<li>Select \u201cExpReg\u201d from the STAT then CALC menu.<\/li>\n \t<li>Use the values returned for <em>a<\/em> and <em>b<\/em> to record the model,[latex]\\,y=a{b}^{x}.[\/latex]<\/li>\n<\/ol>\n<\/li>\n \t<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_08_01\" class=\"textbox examples\">\n<div id=\"fs-id1647375\">\n<div id=\"fs-id1338736\">\n<h3>Using Exponential Regression to Fit a Model to Data<\/h3>\n<p id=\"fs-id1424549\">In 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from 2,871 crashes were used to measure the association of a person\u2019s blood alcohol level (BAC) with the risk of being in an accident. <a class=\"autogenerated-content\" href=\"#Table_04_08_01\">(Figure)<\/a> shows results from the study[footnote]\u2022Source: Indiana University Center for Studies of Law in Action, 2007[\/footnote] . The <em>relative risk<\/em> is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of 0.09 is 3.54 times as likely to crash as a person who has not been drinking alcohol.<\/p>\n\n<table id=\"Table_04_08_01\" summary=\"Two rows and thirteen columns. The first row is labeled, \u201cBAC\u201d, and the second row is labeled, \u201cRelative Risk of Crashing\u201d. Reading the columns as ordered pairs, we have the following values: (0, 1), (0.01, 1.03), (0.03, 1.06), (0.05, 1.38), (0.07, 2.09), (0.09, 3.54), (0.11, 6.41), (0.13, 12.6), (0.15, 22.1), (0.17, 39.05), (0.19, 65.32), and (0.21, 4.394).\">\n<tbody>\n<tr>\n<td><strong>BAC<\/strong><\/td>\n<td>0<\/td>\n<td>0.01<\/td>\n<td>0.03<\/td>\n<td>0.05<\/td>\n<td>0.07<\/td>\n<td>0.09<\/td>\n<\/tr>\n<tr>\n<td><strong>Relative Risk of Crashing<\/strong><\/td>\n<td>1<\/td>\n<td>1.03<\/td>\n<td>1.06<\/td>\n<td>1.38<\/td>\n<td>2.09<\/td>\n<td>3.54<\/td>\n<\/tr>\n<tr>\n<td><strong>BAC<\/strong><\/td>\n<td>0.11<\/td>\n<td>0.13<\/td>\n<td>0.15<\/td>\n<td>0.17<\/td>\n<td>0.19<\/td>\n<td>0.21<\/td>\n<\/tr>\n<tr>\n<td><strong>Relative Risk of Crashing<\/strong><\/td>\n<td>6.41<\/td>\n<td>12.6<\/td>\n<td>22.1<\/td>\n<td>39.05<\/td>\n<td>65.32<\/td>\n<td>99.78<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1326068\" type=\"a\">\n \t<li>Let[latex]\\,x\\,[\/latex]represent the BAC level, and let[latex]\\,y\\,[\/latex]represent the corresponding relative risk. Use exponential regression to fit a model to these data.<\/li>\n \t<li>After 6 drinks, a person weighing 160 pounds will have a BAC of about[latex]\\,0.16.\\,[\/latex]How many times more likely is a person with this weight to crash if they drive after having a 6-pack of beer? Round to the nearest hundredth.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1588497\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1588497\"]\n<ol id=\"fs-id1588497\" type=\"a\">\n \t<li>Using the STAT then EDIT menu on a graphing utility, list the BAC values in L1 and the relative risk values in L2. Then use the STATPLOT feature to verify that the scatterplot follows the exponential pattern shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_001\">(Figure)<\/a>:\n<div id=\"CNX_Precalc_Figure_04_08_001\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141204\/CNX_Precalc_Figure_04_08_001.jpg\" alt=\"Graph of a scattered plot.\" width=\"487\" height=\"475\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1300931\">Use the \u201cExpReg\u201d command from the STAT then CALC menu to obtain the exponential model,<\/p>\n\n<div id=\"eip-id1165134361347\" class=\"unnumbered\">[latex]y=0.58304829{\\left(2.20720213\\text{E}10\\right)}^{x}[\/latex]<\/div>\n<p id=\"fs-id1370824\">Converting from scientific notation, we have:<\/p>\n\n<div id=\"eip-id1165134129943\" class=\"unnumbered\">[latex]y=0.58304829{\\left(\\text{22,072,021,300}\\right)}^{x}[\/latex]<\/div>\n<p id=\"fs-id1327158\">Notice that[latex]\\,{r}^{2}\\approx 0.97\\,[\/latex]which indicates the model is a good fit to the data. To see this, graph the model in the same window as the scatterplot to verify it is a good fit as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_002\">(Figure)<\/a>:<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_08_002\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141206\/CNX_Precalc_Figure_04_08_002.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"487\" height=\"475\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div><\/li>\n \t<li>\n<p id=\"fs-id1598603\">Use the model to estimate the risk associated with a BAC of[latex]\\,0.16.\\,[\/latex]Substitute[latex]\\,0.16\\,[\/latex]for[latex]\\,x\\,[\/latex]in the model and solve for[latex]\\,y.[\/latex]<\/p>\n\n<div id=\"eip-id1165137430511\" class=\"unnumbered\">[latex]\\begin{array}{lll}y\\hfill &amp; =0.58304829{\\left(\\text{22,072,021,300}\\right)}^{x}\\hfill &amp; \\text{Use the regression model found in part (a)}\\text{.}\\hfill \\\\ \\hfill &amp; =0.58304829{\\left(\\text{22,072,021,300}\\right)}^{0.16}\\hfill &amp; \\text{Substitute 0}\\text{.16 for }x\\text{.}\\hfill \\\\ \\hfill &amp; \\approx \\text{26}\\text{.35}\\hfill &amp; \\text{Round to the nearest hundredth}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\nIf a 160-pound person drives after having 6 drinks, he or she is about 26.35 times more likely to crash than if driving while sober.[\/hidden-answer]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1530201\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_08_01\">\n<div id=\"fs-id1433554\">\n<p id=\"fs-id1345853\"><a class=\"autogenerated-content\" href=\"#Table_04_08_02\">(Figure)<\/a> shows a recent graduate\u2019s credit card balance each month after graduation.<\/p>\n\n<table id=\"Table_04_08_02\" summary=\"Two rows and ten columns. The first row is labeled, \u201cMonth\u201d, and the second row is labeled, \u201cDebt (\ud83d\udcb2)\u201d. Reading the columns as ordered pairs, we have the following values: (1, 620.00), (2, 761.88), (3, 899.80), (4, 1039.93), (5, 1270.63), (6, 1589.04), (7, 1851.31), and (8, 2154.92).\">\n<tbody>\n<tr>\n<td><strong>Month<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>Debt ($)<\/strong><\/td>\n<td>620.00<\/td>\n<td>761.88<\/td>\n<td>899.80<\/td>\n<td>1039.93<\/td>\n<td>1270.63<\/td>\n<td>1589.04<\/td>\n<td>1851.31<\/td>\n<td>2154.92<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1420469\" type=\"a\">\n \t<li>Use exponential regression to fit a model to these data.<\/li>\n \t<li>If spending continues at this rate, what will the graduate\u2019s credit card debt be one year after graduating?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1424097\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1424097\"]\n<ol id=\"fs-id1424097\" type=\"a\">\n \t<li>The exponential regression model that fits these data is[latex]\\,y=522.88585984{\\left(1.19645256\\right)}^{x}.[\/latex]<\/li>\n \t<li>If spending continues at this rate, the graduate\u2019s credit card debt will be $4,499.38 after one year.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1395488\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1950428\"><strong>Is it reasonable to assume that an exponential regression model will represent a situation indefinitely?<\/strong><\/p>\n<p id=\"fs-id1693939\"><em>No. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation).<\/em><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1137429\" class=\"bc-section section\">\n<h3>Building a Logarithmic Model from Data<\/h3>\n<p id=\"fs-id1638611\">Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the <em>way<\/em> they increase or decrease that helps us determine whether a <span class=\"no-emphasis\">logarithmic model<\/span> is best.<\/p>\n<p id=\"fs-id1294851\">Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we\u2019ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic <span class=\"no-emphasis\">regression analysis<\/span>, we use the form of the logarithmic function most commonly used on graphing utilities,[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right).\\,[\/latex]For this function<\/p>\n\n<ul id=\"fs-id1505796\">\n \t<li>All input values,[latex]\\,x,[\/latex]must be greater than zero.<\/li>\n \t<li>The point[latex]\\,\\left(1,a\\right)\\,[\/latex]is on the graph of the model.<\/li>\n \t<li>If[latex]\\,b&gt;0,[\/latex]the model is increasing. Growth increases rapidly at first and then steadily slows over time.<\/li>\n \t<li>If[latex]\\,b&lt;0,[\/latex]the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.<\/li>\n<\/ul>\n<div id=\"fs-id1675578\" class=\"textbox key-takeaways\">\n<h3>Logarithmic Regression<\/h3>\n<p id=\"fs-id882689\"><em>Logarithmic regression<\/em> is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command \u201cLnReg\u201d on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form,<\/p>\n\n<div id=\"eip-id1165132974342\" class=\"unnumbered\">[latex]y=a+b\\mathrm{ln}\\left(x\\right)[\/latex]<\/div>\n<p id=\"fs-id1638058\">Note that<\/p>\n\n<ul id=\"fs-id934921\">\n \t<li>all input values,[latex]\\,x,[\/latex]must be non-negative.<\/li>\n \t<li>when[latex]\\,b&gt;0,[\/latex]the model is increasing.<\/li>\n \t<li>when[latex]\\,b&lt;0,[\/latex]the model is decreasing.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1530388\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1395706\"><strong>Given a set of data, perform logarithmic regression using a graphing utility.<\/strong><\/p>\n\n<ol id=\"fs-id1616209\" type=\"1\">\n \t<li>Use the STAT then EDIT menu to enter given data.\n<ol id=\"fs-id1528962\" type=\"a\">\n \t<li>Clear any existing data from the lists.<\/li>\n \t<li>List the input values in the L1 column.<\/li>\n \t<li>List the output values in the L2 column.<\/li>\n<\/ol>\n<\/li>\n \t<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\n<ol id=\"fs-id882304\" type=\"a\">\n \t<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\n \t<li>Verify the data follow a logarithmic pattern.<\/li>\n<\/ol>\n<\/li>\n \t<li>Find the equation that models the data.\n<ol id=\"fs-id1107843\" type=\"a\">\n \t<li>Select \u201cLnReg\u201d from the STAT then CALC menu.<\/li>\n \t<li>Use the values returned for <em>a<\/em> and <em>b<\/em> to record the model,[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right).[\/latex]<\/li>\n<\/ol>\n<\/li>\n \t<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_08_02\" class=\"textbox examples\">\n<div id=\"fs-id1616172\">\n<div id=\"fs-id1586202\">\n<h3>Using Logarithmic Regression to Fit a Model to Data<\/h3>\n<p id=\"fs-id1675089\">Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20th century.<\/p>\n<p id=\"eip-id1165134068998\"><a class=\"autogenerated-content\" href=\"#Table_04_08_03\">(Figure)<\/a> shows the average life expectancies, in years, of Americans from 1900\u20132010[footnote]\u2022Source: Center for Disease Control and Prevention, 2013[\/footnote] .<\/p>\n\n<table id=\"Table_04_08_03\" summary=\"Two rows and twelve columns. The first row is labeled, \u201cYear\u201d, and the second row is labeled, \u201cLife Expectancy (Years)\u201d. Reading the columns as ordered pairs, we have the following values: (1900, 47.3), (1910, 50.0), (1920, 54.1), (1930, 59.7), (1940, 62.9), (1950, 68.2), (1960, 69.7), (1970, 70.8), (1980, 73,7), (1990, 75.4), (2000, 76.8) and (2010, 78.7).\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>1900<\/td>\n<td>1910<\/td>\n<td>1920<\/td>\n<td>1930<\/td>\n<td>1940<\/td>\n<td>1950<\/td>\n<\/tr>\n<tr>\n<td><strong>Life Expectancy(Years)<\/strong><\/td>\n<td>47.3<\/td>\n<td>50.0<\/td>\n<td>54.1<\/td>\n<td>59.7<\/td>\n<td>62.9<\/td>\n<td>68.2<\/td>\n<\/tr>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>1960<\/td>\n<td>1970<\/td>\n<td>1980<\/td>\n<td>1990<\/td>\n<td>2000<\/td>\n<td>2010<\/td>\n<\/tr>\n<tr>\n<td><strong>Life Expectancy(Years)<\/strong><\/td>\n<td>69.7<\/td>\n<td>70.8<\/td>\n<td>73.7<\/td>\n<td>75.4<\/td>\n<td>76.8<\/td>\n<td>78.7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id899809\" type=\"a\">\n \t<li>Let[latex]\\,x\\,[\/latex]represent time in decades starting with[latex]\\,x=1\\,[\/latex]for the year 1900,[latex]\\,x=2\\,[\/latex]for the year 1910, and so on. Let[latex]\\,y\\,[\/latex]represent the corresponding life expectancy. Use logarithmic regression to fit a model to these data.<\/li>\n \t<li>Use the model to predict the average American life expectancy for the year 2030.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1601326\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1601326\"]\n<ol id=\"fs-id1601326\" type=\"a\">\n \t<li>Using the STAT then EDIT menu on a graphing utility, list the years using values 1\u201312 in L1 and the corresponding life expectancy in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logarithmic pattern as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_003\">(Figure)<\/a>:\n<div id=\"CNX_Precalc_Figure_04_08_003\" class=\"medium aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141215\/CNX_Precalc_Figure_04_08_003.jpg\" alt=\"Graph of a scattered plot.\" width=\"731\" height=\"437\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1381567\">Use the \u201cLnReg\u201d command from the STAT then CALC menu to obtain the logarithmic model,<\/p>\n\n<div id=\"eip-id1165137898768\" class=\"unnumbered\">[latex]y=42.52722583+13.85752327\\mathrm{ln}\\left(x\\right)[\/latex]<\/div>\n<p id=\"fs-id1677824\">Next, graph the model in the same window as the scatterplot to verify it is a good fit as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_004\">(Figure)<\/a>:<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_08_004\" class=\"medium aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141222\/CNX_Precalc_Figure_04_08_004.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"731\" height=\"440\"> <strong>Figure 4.<\/strong>[\/caption]\n\n<\/div><\/li>\n \t<li>To predict the life expectancy of an American in the year 2030, substitute[latex]\\,x=14\\,[\/latex]for the in the model and solve for[latex]\\,y:[\/latex]\n<div id=\"eip-id1165132035969\" class=\"unnumbered\">[latex]\\begin{array}{lll}y\\hfill &amp; =42.52722583+13.85752327\\mathrm{ln}\\left(x\\right)\\hfill &amp; \\text{Use the regression model found in part (a)}\\text{.}\\hfill \\\\ \\hfill &amp; =42.52722583+13.85752327\\mathrm{ln}\\left(14\\right)\\hfill &amp; \\text{Substitute 14 for }x\\text{.}\\hfill \\\\ \\hfill &amp; \\approx \\text{79}\\text{.1}\\hfill &amp; \\text{Round to the nearest tenth.}\\hfill \\end{array}[\/latex]<\/div>\nIf life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the year 2030.[\/hidden-answer]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1338539\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_08_02\">\n<div id=\"fs-id1527553\">\n<p id=\"fs-id899893\">Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. <a class=\"autogenerated-content\" href=\"#Table_04_08_04\">(Figure)<\/a> shows the number of games sold, in thousands, from the years 2000\u20132010.<\/p>\n\n<table id=\"Table_04_08_04\" summary=\"Two rows and twelve columns. The first row is labeled, \u201cYear\u201d, and the second row is labeled, \u201cNumber Sold (Thousands)\u201d. Reading the columns as ordered pairs, we have the following values: (2000, 142), (2001, 149), (2002, 154), (2003, 155), (2004, 159), (2005, 161), (2006, 163), (2007, 164), (2008, 164), (2009, 166), and (2010, 167).\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2000<\/td>\n<td>2001<\/td>\n<td>2002<\/td>\n<td>2003<\/td>\n<td>2004<\/td>\n<td>2005<\/td>\n<\/tr>\n<tr>\n<td><strong>Number Sold (thousands)<\/strong><\/td>\n<td>142<\/td>\n<td>149<\/td>\n<td>154<\/td>\n<td>155<\/td>\n<td>159<\/td>\n<td>161<\/td>\n<\/tr>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>-<\/td>\n<\/tr>\n<tr>\n<td><strong>Number Sold (thousands)<\/strong><\/td>\n<td>163<\/td>\n<td>164<\/td>\n<td>164<\/td>\n<td>166<\/td>\n<td>167<\/td>\n<td>-<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1420568\" type=\"a\">\n \t<li>Let[latex]\\,x\\,[\/latex]represent time in years starting with[latex]\\,x=1\\,[\/latex]for the year 2000. Let[latex]\\,y\\,[\/latex]represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data.<\/li>\n \t<li>If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1365229\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1365229\"]\n<ol id=\"fs-id1365229\" type=\"a\">\n \t<li>The logarithmic regression model that fits these data is[latex]\\,y=141.91242949+10.45366573\\mathrm{ln}\\left(x\\right)\\,[\/latex]<\/li>\n \t<li>If sales continue at this rate, about 171,000 games will be sold in the year 2015.[\/hidden-answer]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1599403\" class=\"bc-section section\">\n<h3>Building a Logistic Model from Data<\/h3>\n<p id=\"fs-id1316523\">Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or <em>limiting value<\/em>. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.<\/p>\n<p id=\"fs-id1677718\">It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic <span class=\"no-emphasis\">regression analysis<\/span>, we use the form most commonly used on graphing utilities:<\/p>\n\n<div id=\"eip-154\" class=\"unnumbered aligncenter\">[latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex]<\/div>\n<p id=\"fs-id1310104\">Recall that:<\/p>\n\n<ul id=\"fs-id1294720\">\n \t<li>[latex]\\frac{c}{1+a}\\,[\/latex]is the initial value of the model.<\/li>\n \t<li>when[latex]\\,b&gt;0,[\/latex] the model increases rapidly at first until it reaches its point of maximum growth rate,[latex]\\,\\left(\\frac{\\mathrm{ln}\\left(a\\right)}{b},\\frac{c}{2}\\right).\\,[\/latex]At that point, growth steadily slows and the function becomes asymptotic to the upper bound[latex]\\,y=c.[\/latex]<\/li>\n \t<li>[latex]c\\,[\/latex]\nis the limiting value, sometimes called the <em>carrying capacity<\/em>, of the model.<\/li>\n<\/ul>\n<div id=\"fs-id1454974\" class=\"textbox key-takeaways\">\n<h3>Logistic Regression<\/h3>\n<p id=\"fs-id1583226\"><em>Logistic regression<\/em> is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command \u201cLogistic\u201d on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form<\/p>\n\n<div class=\"unnumbered\">[latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex]<\/div>\n<p id=\"fs-id1701600\">Note that<\/p>\n\n<ul id=\"fs-id1358262\">\n \t<li>The initial value of the model is[latex]\\,\\frac{c}{1+a}.[\/latex]<\/li>\n \t<li>Output values for the model grow closer and closer to[latex]\\,y=c\\,[\/latex]as time increases.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1361145\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1676582\"><strong>Given a set of data, perform logistic regression using a graphing utility.<\/strong><\/p>\n\n<ol id=\"fs-id1690756\" type=\"1\">\n \t<li>Use the STAT then EDIT menu to enter given data.\n<ol id=\"fs-id1562359\" type=\"a\">\n \t<li>Clear any existing data from the lists.<\/li>\n \t<li>List the input values in the L1 column.<\/li>\n \t<li>List the output values in the L2 column.<\/li>\n<\/ol>\n<\/li>\n \t<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\n<ol id=\"fs-id1431005\" type=\"a\">\n \t<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\n \t<li>Verify the data follow a logistic pattern.<\/li>\n<\/ol>\n<\/li>\n \t<li>Find the equation that models the data.\n<ol id=\"fs-id1585412\" type=\"a\">\n \t<li>Select \u201cLogistic\u201d from the STAT then CALC menu.<\/li>\n \t<li>Use the values returned for[latex]\\,a,[\/latex][latex]\\,b,[\/latex] and[latex]\\,c\\,[\/latex]to record the model,[latex]\\,y=\\frac{c}{1+a{e}^{-bx}}.[\/latex]<\/li>\n<\/ol>\n<\/li>\n \t<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_08_03\" class=\"textbox examples\">\n<div id=\"fs-id1646727\">\n<div id=\"fs-id1523395\">\n<h3>Using Logistic Regression to Fit a Model to Data<\/h3>\n<p id=\"fs-id1422087\">Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. <a class=\"autogenerated-content\" href=\"#Table_04_08_05\">(Figure)<\/a> shows the percentage of Americans with cellular service between the years 1995 and 2012[footnote]\u2022Source: The World Bank, 2013[\/footnote] .<\/p>\n\n<table id=\"Table_04_08_05\" summary=\"Nineteen rows and two columns. The first column is labeled, \u201cYear\u201d, and the second column is labeled, \u201cAmericans with Cellular Service (%)\u201d. Reading the columns as ordered pairs, we have the following values: (1995, 12.69), (1996, 16.35), (1997, 20.29), (1998, 25.08), (1999, 30.81), (2000, 38.75), (2001, 45.00), (2002, 49.16), (2003, 55.15), (2004, 62.85), (2005, 68.63), (2006, 76.64), (2007, 82.47), (2008, 85.68), (2009, 89.14), (2010, 91.86), (2011, 95.28), and (2012, 98.17).\">\n<thead>\n<tr>\n<th>Year<\/th>\n<th>Americans with Cellular Service (%)<\/th>\n<th>Year<\/th>\n<th>Americans with Cellular Service (%)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1995<\/td>\n<td>12.69<\/td>\n<td>2004<\/td>\n<td>62.852<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>16.35<\/td>\n<td>2005<\/td>\n<td>68.63<\/td>\n<\/tr>\n<tr>\n<td>1997<\/td>\n<td>20.29<\/td>\n<td>2006<\/td>\n<td>76.64<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>25.08<\/td>\n<td>2007<\/td>\n<td>82.47<\/td>\n<\/tr>\n<tr>\n<td>1999<\/td>\n<td>30.81<\/td>\n<td>2008<\/td>\n<td>85.68<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>38.75<\/td>\n<td>2009<\/td>\n<td>89.14<\/td>\n<\/tr>\n<tr>\n<td>2001<\/td>\n<td>45.00<\/td>\n<td>2010<\/td>\n<td>91.86<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>49.16<\/td>\n<td>2011<\/td>\n<td>95.28<\/td>\n<\/tr>\n<tr>\n<td>2003<\/td>\n<td>55.15<\/td>\n<td>2012<\/td>\n<td>98.17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1624806\" type=\"a\">\n \t<li>Let[latex]\\,x\\,[\/latex]represent time in years starting with[latex]\\,x=0\\,[\/latex]for the year 1995. Let[latex]\\,y\\,[\/latex]represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data.<\/li>\n \t<li>Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent.<\/li>\n \t<li>Discuss the value returned for the upper limit,[latex]\\,c.\\,[\/latex]What does this tell you about the model? What would the limiting value be if the model were exact?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1455761\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1455761\"]\n<ol id=\"fs-id1455761\" type=\"a\">\n \t<li>Using the STAT then EDIT menu on a graphing utility, list the years using values 0\u201315 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_005\">(Figure)<\/a>:\n<div id=\"CNX_Precalc_Figure_04_08_005\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141225\/CNX_Precalc_Figure_04_08_005.jpg\" alt=\"Graph of a scattered plot.\" width=\"975\" height=\"479\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1366003\">Use the \u201cLogistic\u201d command from the STAT then CALC menu to obtain the logistic model,<\/p>\n\n<div id=\"eip-id1165134104896\" class=\"unnumbered\">[latex]y=\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}[\/latex]<\/div>\n<p id=\"fs-id1650001\">Next, graph the model in the same window as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_006\">(Figure)<\/a> the scatterplot to verify it is a good fit:<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_08_006\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141228\/CNX_Precalc_Figure_04_08_006.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"975\" height=\"479\"> <strong>Figure 6.<\/strong>[\/caption]\n\n<\/div><\/li>\n \t<li>\n<p id=\"fs-id1410550\">To approximate the percentage of Americans with cellular service in the year 2013, substitute[latex]\\,x=18\\,[\/latex]for the in the model and solve for[latex]\\,y:[\/latex]<\/p>\n\n<div id=\"eip-9\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{lll}y\\hfill &amp; =\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}\\hfill &amp; \\text{Use the regression model found in part (a)}.\\hfill \\\\ \\hfill &amp; =\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013\\left(18\\right)}}\\hfill &amp; \\text{Substitute 18 for }x.\\hfill \\\\ \\hfill &amp; \\approx \\text{99}\\text{.3 }\\hfill &amp; \\text{Round to the nearest tenth}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1569617\">According to the model, about 98.8% of Americans had cellular service in 2013.<\/p>\n<\/li>\n \t<li>The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be[latex]\\,c=100\\,[\/latex]and the model\u2019s outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service![\/hidden-answer]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1394388\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_08_03\">\n<div id=\"fs-id1521431\">\n<p id=\"fs-id1638072\"><a class=\"autogenerated-content\" href=\"#Table_04_08_06\">(Figure)<\/a> shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012.<\/p>\n\n<table id=\"Table_04_08_06\" summary=\"Seventeen rows and two columns. The first column is labeled, \u201cYear\u201d, and the second column is labeled, \u201cSeal Population (Thousands)\u201d. Reading the columns as ordered pairs, we have the following values: (1997, 3.493), (1998, 5.282), (1999, 6.357), (2000, 9.201), (2001, 11.224), (2002, 12.964), (2003, 16.226), (2004, 18.137), (2005, 19.590), (2006, 21.955), (2007, 22.862), (2008, 23.869), (2009, 24.243), (2010, 24.344), (2011, 24.919), and (2012, 25.108).\">\n<thead>\n<tr>\n<th>Year<\/th>\n<th>Seal Population (Thousands)<\/th>\n<th>Year<\/th>\n<th>Seal Population (Thousands)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1997<\/td>\n<td>3.493<\/td>\n<td>2005<\/td>\n<td>19.590<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>5.282<\/td>\n<td>2006<\/td>\n<td>21.955<\/td>\n<\/tr>\n<tr>\n<td>1999<\/td>\n<td>6.357<\/td>\n<td>2007<\/td>\n<td>22.862<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>9.201<\/td>\n<td>2008<\/td>\n<td>23.869<\/td>\n<\/tr>\n<tr>\n<td>2001<\/td>\n<td>11.224<\/td>\n<td>2009<\/td>\n<td>24.243<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>12.964<\/td>\n<td>2010<\/td>\n<td>24.344<\/td>\n<\/tr>\n<tr>\n<td>2003<\/td>\n<td>16.226<\/td>\n<td>2011<\/td>\n<td>24.919<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>18.137<\/td>\n<td>2012<\/td>\n<td>25.108<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1366862\" type=\"a\">\n \t<li>Let[latex]\\,x\\,[\/latex]represent time in years starting with[latex]\\,x=0\\,[\/latex]for the year 1997. Let[latex]\\,y\\,[\/latex]represent the number of seals in thousands. Use logistic regression to fit a model to these data.<\/li>\n \t<li>Use the model to predict the seal population for the year 2020.<\/li>\n \t<li>To the nearest whole number, what is the limiting value of this model?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1609961\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1609961\"]\n<ol id=\"fs-id1609961\" type=\"a\">\n \t<li>The logistic regression model that fits these data is[latex]\\,y=\\frac{25.65665979}{1+6.113686306{e}^{-0.3852149008x}}.[\/latex]<\/li>\n \t<li>If the population continues to grow at this rate, there will be about[latex]\\,\\text{25,634}\\,[\/latex]seals in 2020.<\/li>\n \t<li>To the nearest whole number, the carrying capacity is 25,657.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1384374\" class=\"precalculus media\">\n<p id=\"fs-id1676985\">Access this online resource for additional instruction and practice with exponential function models.<\/p>\n\n<ul id=\"fs-id1676988\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/pregresscalc\">Exponential Regression on a Calculator<\/a><\/li>\n<\/ul>\n<\/div>\n<p id=\"eip-136\">Visit <a href=\"http:\/\/openstaxcollege.org\/l\/PreCalcLPC04\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n\n<\/div>\n<div id=\"fs-id1294229\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1410554\">\n \t<li>Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.<\/li>\n \t<li>We use the command \u201cExpReg\u201d on a graphing utility to fit function of the form[latex]\\,y=a{b}^{x}\\,[\/latex]to a set of data points. See <a class=\"autogenerated-content\" href=\"#Example_04_08_01\">(Figure)<\/a>.<\/li>\n \t<li>Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.<\/li>\n \t<li>We use the command \u201cLnReg\u201d on a graphing utility to fit a function of the form[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right)\\,[\/latex]to a set of data points. See <a class=\"autogenerated-content\" href=\"#Example_04_08_02\">(Figure)<\/a>.<\/li>\n \t<li>Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.<\/li>\n \t<li>We use the command \u201cLogistic\u201d on a graphing utility to fit a function of the form[latex]\\,y=\\frac{c}{1+a{e}^{-bx}}\\,[\/latex]to a set of data points. See <a class=\"autogenerated-content\" href=\"#Example_04_08_03\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1700714\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1700718\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1402840\">\n<div id=\"fs-id1402843\">\n<p id=\"fs-id1424627\">What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1660264\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1660264\"]\n<p id=\"fs-id1660264\">Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1692586\">\n<div id=\"fs-id1692588\">\n<p id=\"fs-id1033498\">What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1447042\">\n<div id=\"fs-id1523478\">\n<p id=\"fs-id1523480\">What is regression analysis? Describe the process of performing regression analysis on a graphing utility.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1648021\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1648021\"]\n<p id=\"fs-id1648021\">Regression analysis is the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the given points using the STAT then EDIT menu. Next graph the scatter plot using the STAT PLOT feature. The shape of the data points on the scatter graph can help determine which regression feature to use. Once this is determined, select the appropriate regression analysis command from the STAT then CALC menu.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1562322\">\n<div id=\"fs-id1562324\">\n<p id=\"fs-id1271600\">What might a scatterplot of data points look like if it were best described by a logarithmic model?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1638051\">\n<div id=\"fs-id1638053\">\n<p id=\"fs-id916950\">What does the <em>y<\/em>-intercept on the graph of a logistic equation correspond to for a population modeled by that equation?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1586837\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1586837\"]\n<p id=\"fs-id1586837\">The <em>y<\/em>-intercept on the graph of a logistic equation corresponds to the initial population for the population model.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1457107\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1404306\">For the following exercises, match the given function of best fit with the appropriate scatterplot in <a class=\"autogenerated-content\" href=\"#CNX_PreCalc_Figure_04_08_201\">(Figure)<\/a> through <a class=\"autogenerated-content\" href=\"#CNX_PreCalc_Figure_04_08_205\">(Figure)<\/a><strong>. <\/strong>Answer using the letter beneath the matching graph.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_08_201\" class=\"small aligncenter\"><span id=\"fs-id299196\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141230\/CNX_PreCalc_Figure_04_08_201.jpg\" alt=\"Graph of a scattered plot.\"><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_04_08_202\" class=\"small aligncenter\"><span id=\"fs-id1425423\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141232\/CNX_PreCalc_Figure_04_08_202.jpg\" alt=\"Graph of a scattered plot.\"><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_04_08_203\" class=\"small aligncenter\"><span id=\"fs-id1200734\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141240\/CNX_PreCalc_Figure_04_08_203.jpg\" alt=\"Graph of a scattered plot.\"><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_04_08_204\" class=\"small aligncenter\"><span id=\"fs-id1692902\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141243\/CNX_PreCalc_Figure_04_08_204.jpg\" alt=\"Graph of a scattered plot.\"><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_04_08_205\" class=\"small aligncenter\"><span id=\"fs-id1425975\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141253\/CNX_PreCalc_Figure_04_08_205.jpg\" alt=\"Graph of a scattered plot.\"><\/span><\/div>\n<div id=\"fs-id1670879\">\n<div id=\"fs-id1670881\">\n<p id=\"fs-id1700443\">[latex]y=10.209{e}^{-0.294x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1701699\">\n<div id=\"fs-id1701701\">\n<p id=\"fs-id1428638\">[latex]y=5.598-1.912\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1423720\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1423720\"]\n<p id=\"fs-id1423720\">C<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1519709\">\n<div id=\"fs-id1519711\">\n<p id=\"fs-id1422545\">[latex]y=2.104{\\left(1.479\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1693326\">\n<div id=\"fs-id1692337\">\n<p id=\"fs-id1692340\">[latex]y=4.607+2.733\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1696917\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1696917\"]\n<p id=\"fs-id1696917\">B<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1410055\">\n<div id=\"fs-id1385282\">\n<p id=\"fs-id1385284\">[latex]y=\\frac{14.005}{1+2.79{e}^{-0.812x}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1659453\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<div id=\"fs-id1522155\">\n<div id=\"fs-id1522157\">\n<p id=\"fs-id1522159\">To the nearest whole number, what is the initial value of a population modeled by the logistic equation[latex]\\,P\\left(t\\right)=\\frac{175}{1+6.995{e}^{-0.68t}}?\\,[\/latex]What is the carrying capacity?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1661536\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1661536\"]\n<p id=\"fs-id1661536\">[latex]P\\left(0\\right)=22\\,[\/latex]; 175<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1346082\">\n<div id=\"fs-id1346085\">\n<p id=\"fs-id1660230\">Rewrite the exponential model[latex]\\,A\\left(t\\right)=1550{\\left(1.085\\right)}^{x}\\,[\/latex]as an equivalent model with base[latex]\\,e.\\,[\/latex]Express the exponent to four significant digits.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1697446\">\n<div id=\"fs-id1675823\">\n<p id=\"fs-id1675825\">A logarithmic model is given by the equation[latex]\\,h\\left(p\\right)=67.682-5.792\\mathrm{ln}\\left(p\\right).\\,[\/latex]To the nearest hundredth, for what value of[latex]\\,p\\,[\/latex]does[latex]\\,h\\left(p\\right)=62?[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1086031\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1086031\"]\n<p id=\"fs-id1086031\">[latex]p\\approx 2.67[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405194\">\n<div id=\"fs-id1405196\">\n<p id=\"fs-id1381601\">A logistic model is given by the equation[latex]\\,P\\left(t\\right)=\\frac{90}{1+5{e}^{-0.42t}}.\\,[\/latex]To the nearest hundredth, for what value of <em>t<\/em> does[latex]\\,P\\left(t\\right)=45?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1439754\">\n<div id=\"fs-id1439756\">\n<p id=\"fs-id1569418\">What is the <em>y<\/em>-intercept on the graph of the logistic model given in the previous exercise?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1697176\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1697176\"]\n<p id=\"fs-id1697176\"><em>y<\/em>-intercept:[latex]\\,\\left(0,15\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1702994\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1422233\">For the following exercises, use this scenario: The population[latex]\\,P\\,[\/latex]of a koi pond over[latex]\\,x\\,[\/latex]months is modeled by the function[latex]\\,P\\left(x\\right)=\\frac{68}{1+16{e}^{-0.28x}}.[\/latex]<\/p>\n\n<div id=\"fs-id1246586\">\n<div id=\"fs-id1246589\">\n<p id=\"fs-id1457169\">Graph the population model to show the population over a span of[latex]\\,3\\,[\/latex]years.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1408601\">\n<div id=\"fs-id1408603\">\n<p id=\"fs-id1408605\">What was the initial population of koi?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1440059\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1440059\"]\n<p id=\"fs-id1440059\">[latex]4\\,[\/latex]koi<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1535423\">\n<div id=\"fs-id1535425\">\n<p id=\"fs-id1678373\">How many koi will the pond have after one and a half years?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1678378\">\n<div id=\"fs-id1423485\">\n<p id=\"fs-id1423487\">How many months will it take before there are[latex]\\,20\\,[\/latex]koi in the pond?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1394674\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1394674\"]\n<p id=\"fs-id1394674\">about[latex]\\,6.8\\,[\/latex]months.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1582861\">\n<div id=\"fs-id1420623\">\n<p id=\"fs-id1420625\">Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity.<\/p>\n\n<\/div>\n<div id=\"eip-id2711495\">\n<div class=\"textbox shaded\">[reveal-answer q=\"647263\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"647263\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141255\/CNX_Precalc_Figure_04_08_207.jpg\" alt=\"Graph of the intersection of P(t)=68\/(1+16e^(-0.28t)) and y=34.\">[\/hidden-answer]<\/div>\n&nbsp;\n\n<\/div>\n<\/div>\n<p id=\"fs-id1706412\">For the following exercises, use this scenario: The population[latex]\\,P\\,[\/latex]of an endangered species habitat for wolves is modeled by the function[latex]\\,P\\left(x\\right)=\\frac{558}{1+54.8{e}^{-0.462x}},[\/latex] where[latex]\\,x\\,[\/latex]is given in years.<\/p>\n\n<div id=\"fs-id1658971\">\n<div id=\"fs-id1658973\">\n<p id=\"fs-id1658975\">Graph the population model to show the population over a span of[latex]\\,10\\,[\/latex]years.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1701908\">\n<div id=\"fs-id1504971\">\n<p id=\"fs-id1504973\">What was the initial population of wolves transported to the habitat?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1409591\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1409591\"]\n<p id=\"fs-id1409591\">[latex]10\\,[\/latex]wolves<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1505494\">\n<div id=\"fs-id1505496\">\n<p id=\"fs-id1505498\">How many wolves will the habitat have after[latex]\\,3\\,[\/latex]years?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1271453\">\n<div id=\"fs-id925282\">\n<p id=\"fs-id925284\">How many years will it take before there are[latex]\\,100\\,[\/latex]wolves in the habitat?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1459669\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1459669\"]\n<p id=\"fs-id1459669\">about 5.4 years.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1601914\">\n<div id=\"fs-id1601916\">\n<p id=\"fs-id1523061\">Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1597787\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_07\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_04_08_07\" summary=\"Two columns and seven row. The first column labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 1125), (2, 1495), (3, 2310), (4, 3295), (5, 4650), and (6, 6361).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>1125<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>1495<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>2310<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>3294<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>4650<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>6361<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1432918\">\n<div id=\"fs-id1699848\">\n<p id=\"fs-id1699850\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1648119\"]Show Solution[\/reveal-answer][hidden-answer a=\"1648119\"]<span id=\"fs-id1803598\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141259\/CNX_PreCalc_Figure_04_08_210.jpg\" alt=\"Graph of the table\u2019s values.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1514386\">\n<div id=\"fs-id1648035\">\n<p id=\"fs-id1648037\">Use the regression feature to find an exponential function that best fits the data in the table.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1586463\">\n<div id=\"fs-id1586466\">\n<p id=\"fs-id1597729\">Write the exponential function as an exponential equation with base[latex]\\,e.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1600227\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1600227\"]\n<p id=\"fs-id1600227\">[latex]f\\left(x\\right)=776.682{e}^{0.3549x}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1447084\">\n<div id=\"fs-id1447087\">\n<p id=\"fs-id1447089\">Graph the exponential equation on the scatter diagram.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1447005\">\n<div id=\"fs-id1376767\">\n<p id=\"fs-id1376769\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which[latex]\\,f\\left(x\\right)=4000.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1221397\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1221397\"]\n<p id=\"fs-id1221397\">When[latex]\\,f\\left(x\\right)=4000,[\/latex][latex]x\\approx 4.6.[\/latex]<\/p>\n<span id=\"fs-id1436308\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141306\/CNX_PreCalc_Figure_04_08_212.jpg\" alt=\"Graph of the intersection of a scattered plot with an estimation line and y=4,000.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1511970\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_08\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_04_08_08\" summary=\"Two columns and seven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 555), (2, 383), (3, 307), (4, 210), (5, 158), and (6, 122).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>555<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>383<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>307<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>210<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>158<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>122<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1407921\">\n<div id=\"fs-id1407923\">\n<p id=\"fs-id1407925\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1671727\">\n<div id=\"fs-id1692964\">\n<p id=\"fs-id1692966\">Use the regression feature to find an exponential function that best fits the data in the table.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1405103\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1405103\"]\n<p id=\"fs-id1405103\">[latex]f\\left(x\\right)=731.92{\\left(0.738\\right)}^{x}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1652039\">\n<div id=\"fs-id1652041\">\n<p id=\"fs-id1652043\">Write the exponential function as an exponential equation with base[latex]\\,e.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1590772\">\n<div id=\"fs-id1513210\">\n<p id=\"fs-id1513213\">Graph the exponential equation on the scatter diagram.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1435902\"]Show Solution[\/reveal-answer][hidden-answer a=\"1435902\"]<span id=\"fs-id1435907\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141309\/CNX_PreCalc_Figure_04_08_214.jpg\" alt=\"Graph of a scattered plot with an estimation line.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1606195\">\n<div id=\"fs-id1424979\">\n<p id=\"fs-id1424981\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which[latex]\\,f\\left(x\\right)=250.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1460953\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_09\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_04_08_09\" summary=\"Two columns and seven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 5.1), (2, 6.3), (3, 7.3), (4, 7.7), (5, 8.1), and (6, 8.6).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>5.1<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>6.3<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>7.3<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>7.7<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>8.1<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>8.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1371063\">\n<div id=\"fs-id1371065\">\n<p id=\"fs-id1371067\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1380157\"]Show Solution[\/reveal-answer][hidden-answer a=\"1380157\"]<span id=\"fs-id1588274\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141312\/CNX_PreCalc_Figure_04_08_216.jpg\" alt=\"Graph of the table\u2019s values.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1650282\">\n<div id=\"fs-id1650285\">\n<p id=\"fs-id1420307\">Use the LOGarithm option of the REGression feature to find a logarithmic function of the form[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right)\\,[\/latex]that best fits the data in the table.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1432678\">\n<div id=\"fs-id1432680\">\n<p id=\"fs-id1356158\">Use the logarithmic function to find the value of the function when[latex]\\,x=10.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1518348\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1518348\"]\n<p id=\"fs-id1518348\">[latex]f\\left(10\\right)\\approx 9.5[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1392056\">\n<div id=\"fs-id1392058\">\n<p id=\"fs-id1392060\">Graph the logarithmic equation on the scatter diagram.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1649251\">\n<div id=\"fs-id1649253\">\n<p id=\"fs-id1355295\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which[latex]\\,f\\left(x\\right)=7.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1434551\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1434551\"]\n<p id=\"fs-id1434551\">When[latex]\\,f\\left(x\\right)=7,[\/latex] [latex]x\\approx 2.7.[\/latex]<\/p>\n<span id=\"fs-id1381504\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141314\/CNX_PreCalc_Figure_04_08_218.jpg\" alt=\"Graph of the intersection of a scattered plot with an estimation line and y=7.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1694168\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_10\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_04_08_10\" summary=\"Two columns and nine nows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 7.5), (2, 6), (3, 5.2), (4, 4.3), (5, 3.9), (6, 3.4), (7, 3.1), and (8, 2.9).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>7.5<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>5.2<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>4.3<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>3.9<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>3.4<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>3.1<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>2.9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1432065\">\n<div id=\"fs-id1432067\">\n<p id=\"fs-id1432069\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1426060\">\n<div id=\"fs-id1426062\">\n<p id=\"fs-id1426064\">Use the LOGarithm option of the REGression feature to find a logarithmic function of the form[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right)\\,[\/latex]that best fits the data in the table.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1393286\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1393286\"]\n<p id=\"fs-id1393286\">[latex]f\\left(x\\right)=7.544-2.268\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1424700\">\n<div id=\"fs-id1601388\">\n<p id=\"fs-id1601390\">Use the logarithmic function to find the value of the function when[latex]\\,x=10.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1649327\">\n<div id=\"fs-id1649329\">\n<p id=\"fs-id1649331\">Graph the logarithmic equation on the scatter diagram.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1364732\"]Show Solution[\/reveal-answer][hidden-answer a=\"1364732\"]<span id=\"fs-id1364738\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141320\/CNX_PreCalc_Figure_04_08_220.jpg\" alt=\"Graph of a scattered plot with an estimation line.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1451593\">\n<div id=\"fs-id1451595\">\n<p id=\"fs-id1211247\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which[latex]\\,f\\left(x\\right)=8.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1661470\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_11\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_04_08_11\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 8.7), (2, 12.3), (3, 15.4), (4, 18.5), (5, 20.7), (6, 22.5), (7, 23.3), (8, 24), (9, 24.6), and (10, 24.8).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>8.7<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>12.3<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>15.4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>18.5<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>20.7<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>22.5<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>23.3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>24<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>24.6<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>24.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1656649\">\n<div id=\"fs-id1407656\">\n<p id=\"fs-id1407658\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1407662\"]Show Solution[\/reveal-answer][hidden-answer a=\"1407662\"]<span id=\"fs-id1439003\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141323\/CNX_PreCalc_Figure_04_08_222.jpg\" alt=\"Graph of the table\u2019s values.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1423807\">\n<div>\n\nUse the LOGISTIC regression option to find a logistic growth model of the form[latex]\\,y=\\frac{c}{1+a{e}^{-bx}}\\,[\/latex]that best fits the data in the table.\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405629\">\n<div id=\"fs-id1405631\">\n<p id=\"fs-id1671038\">Graph the logistic equation on the scatter diagram.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1671043\"]Show Solution[\/reveal-answer][hidden-answer a=\"1671043\"]<span id=\"fs-id1523360\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141326\/CNX_Precalc_Figure_04_08_223.jpg\" alt=\"Graph of a scattered plot with an estimation line.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1535974\">\n<div id=\"fs-id1508700\">\n<p id=\"fs-id1508702\">To the nearest whole number, what is the predicted carrying capacity of the model?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1508707\">\n<div id=\"fs-id1609403\">\n<p id=\"fs-id1609405\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which the model reaches half its carrying capacity.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1370548\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1370548\"]\n<p id=\"fs-id1370548\">When[latex]\\,f\\left(x\\right)=12.5,[\/latex] [latex]x\\approx 2.1.[\/latex]<\/p>\n<span id=\"fs-id1689750\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141339\/CNX_Precalc_Figure_04_08_224.jpg\" alt=\"Graph of the intersection of a scattered plot with an estimation line and y=12.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1433707\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_12\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_04_08_12\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (0, 12), (2, 28.6), (4, 52.8), (5, 70.3), (7, 99.9), (8, 112.5), (10, 125.8), (11, 127.9), (15, 135.1), and (17, 135.9).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>28.6<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>52.8<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>70.3<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>99.9<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>112.5<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>125.8<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>127.9<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>135.1<\/td>\n<\/tr>\n<tr>\n<td>17<\/td>\n<td>135.9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1440074\">\n<div id=\"fs-id1440076\">\n<p id=\"fs-id1440079\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1520217\">\n<div id=\"fs-id1520219\">\n<p id=\"fs-id1520221\">Use the LOGISTIC regression option to find a logistic growth model of the form[latex]\\,y=\\frac{c}{1+a{e}^{-bx}}\\,[\/latex]that best fits the data in the table.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1375601\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1375601\"]\n<p id=\"fs-id1375601\">[latex]f\\left(x\\right)=\\frac{136.068}{1+10.324{e}^{-0.480x}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1705222\">\n<div id=\"fs-id1705224\">\n<p id=\"fs-id1705226\">Graph the logistic equation on the scatter diagram.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1531111\">\n<div id=\"fs-id1531113\">\n<p id=\"fs-id1531115\">To the nearest whole number, what is the predicted carrying capacity of the model?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1405350\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1405350\"]\n<p id=\"fs-id1405350\">about[latex]136[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1406372\">\n<div id=\"fs-id1406374\">\n<p id=\"fs-id1406377\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which the model reaches half its carrying capacity.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1583097\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1583102\">\n<div id=\"fs-id1583104\">\n<p id=\"fs-id1404836\">Recall that the general form of a logistic equation for a population is given by[latex]\\,P\\left(t\\right)=\\frac{c}{1+a{e}^{-bt}},[\/latex] such that the initial population at time[latex]\\,t=0\\,[\/latex]is[latex]\\,P\\left(0\\right)={P}_{0}.\\,[\/latex]Show algebraically that[latex]\\,\\frac{c-P\\left(t\\right)}{P\\left(t\\right)}=\\frac{c-{P}_{0}}{{P}_{0}}{e}^{-bt}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1366138\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1366138\"]\n<p id=\"fs-id1366138\">Working with the left side of the equation, we see that it can be rewritten as[latex]\\,a{e}^{-bt}:[\/latex]<\/p>\n<p id=\"fs-id1457268\">[latex]\\frac{c-P\\left(t\\right)}{P\\left(t\\right)}=\\frac{c-\\frac{c}{1+a{e}^{-bt}}}{\\frac{c}{1+a{e}^{-bt}}}=\\frac{\\frac{c\\left(1+a{e}^{-bt}\\right)-c}{1+a{e}^{-bt}}}{\\frac{c}{1+a{e}^{-bt}}}=\\frac{\\frac{c\\left(1+a{e}^{-bt}-1\\right)}{1+a{e}^{-bt}}}{\\frac{c}{1+a{e}^{-bt}}}=1+a{e}^{-bt}-1=a{e}^{-bt}[\/latex]<\/p>\n<p id=\"fs-id1460591\">Working with the right side of the equation we show that it can also be rewritten as[latex]\\,a{e}^{-bt}.\\,[\/latex]But first note that when[latex]\\,t=0,[\/latex]\n[latex]\\,{P}_{0}=\\frac{c}{1+a{e}^{-b\\left(0\\right)}}=\\frac{c}{1+a}.\\,[\/latex]Therefore,<\/p>\n<p id=\"fs-id1283132\">[latex]\\frac{c-{P}_{0}}{{P}_{0}}{e}^{-bt}=\\frac{c-\\frac{c}{1+a}}{\\frac{c}{1+a}}{e}^{-bt}=\\frac{\\frac{c\\left(1+a\\right)-c}{1+a}}{\\frac{c}{1+a}}{e}^{-bt}=\\frac{\\frac{c\\left(1+a-1\\right)}{1+a}}{\\frac{c}{1+a}}{e}^{-bt}=\\left(1+a-1\\right){e}^{-bt}=a{e}^{-bt}[\/latex]<\/p>\nThus,[latex]\\,\\frac{c-P\\left(t\\right)}{P\\left(t\\right)}=\\frac{c-{P}_{0}}{{P}_{0}}{e}^{-bt}.[\/latex][\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1435401\">\n<div id=\"fs-id1435404\">\n<p id=\"fs-id1658819\">Use a graphing utility to find an exponential regression formula[latex]\\,f\\left(x\\right)\\,[\/latex]and a logarithmic regression formula[latex]\\,g\\left(x\\right)\\,[\/latex]for the points[latex]\\,\\left(1.5,1.5\\right)\\,[\/latex]and[latex]\\,\\left(8.5,\\text{ 8}\\text{.5}\\right).\\,[\/latex]Round all numbers to 6 decimal places. Graph the points and both formulas along with the line[latex]\\,y=x\\,[\/latex]on the same axis. Make a conjecture about the relationship of the regression formulas.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1583256\">\n<div id=\"fs-id1583258\">\n<p id=\"fs-id1658556\">Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1658563\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1658563\"]\n<p id=\"fs-id1658563\">First rewrite the exponential with base <em>e<\/em>:[latex]\\,f\\left(x\\right)=1.034341{e}^{\\text{0}\\text{.247800x}}.\\,[\/latex]Then test to verify that[latex]\\,f\\left(g\\left(x\\right)\\right)=x,[\/latex]taking rounding error into consideration:<\/p>\n<p id=\"fs-id1513342\">[latex]\\begin{array}{ll}g\\left(f\\left(x\\right)\\right)\\hfill &amp; =4.035510\\mathrm{ln}\\left(1.034341{e}^{\\text{0}\\text{.247800x}}\\,\\right)-0.136259\\hfill \\\\ \\hfill &amp; =4.03551\\left(\\mathrm{ln}\\left(1.034341\\right)+\\mathrm{ln}\\left({e}^{\\text{0}\\text{.2478}x}\\,\\right)\\right)-0.136259\\hfill \\\\ \\hfill &amp; =4.03551\\left(\\mathrm{ln}\\left(1.034341\\right)+\\text{0}\\text{.2478}x\\right)-0.136259\\hfill \\\\ \\hfill &amp; =0.136257+0.999999x-0.136259\\hfill \\\\ \\hfill &amp; =-0.000002+0.999999x\\hfill \\\\ \\hfill &amp; \\approx 0+x\\hfill \\\\ \\hfill &amp; =x\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1700899\">\n<div id=\"fs-id1532222\">\n<p id=\"fs-id1532224\">Find the inverse function[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]for the logistic function[latex]\\,f\\left(x\\right)=\\frac{c}{1+a{e}^{-bx}}.\\,[\/latex]Show all steps.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1631452\">\n<div id=\"fs-id1631454\">\n<p id=\"fs-id1631456\">Use the result from the previous exercise to graph the logistic model[latex]\\,P\\left(t\\right)=\\frac{20}{1+4{e}^{-0.5t}}\\,[\/latex]along with its inverse on the same axis. What are the intercepts and asymptotes of each function?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1440685\"]Show Solution[\/reveal-answer][hidden-answer a=\"1440685\"]<span id=\"fs-id1440690\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141350\/CNX_PreCalc_Figure_04_08_228.jpg\" alt=\"Graph of P(t)=20\/(1+40.5e^(-0.5t)) and P(t)=(ln(4)-ln((20\/t)-1)\/0.5.\"><\/span>\n<p id=\"eip-id1165134486767\">The graph of[latex]\\,P\\left(t\\right)\\,[\/latex]has a <em>y<\/em>-intercept at (0, 4) and horizontal asymptotes at <em>y<\/em> = 0 and <em>y<\/em> = 20. The graph of[latex]\\,{P}^{-1}\\left(t\\right)\\,[\/latex]has an <em>x<\/em>- intercept at (4, 0) and vertical asymptotes at <em>x<\/em> = 0 and <em>x<\/em> = 20.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1582712\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1582715\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/694d2eea-4135-47f3-8c06-60472f7e967c\">Exponential Functions<\/a><\/h4>\n<div id=\"fs-id1705125\">\n<div id=\"fs-id1705127\">\n<p id=\"fs-id1705129\">Determine whether the function[latex]\\,y=156{\\left(0.825\\right)}^{t}\\,[\/latex]represents exponential growth, exponential decay, or neither. Explain<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1504948\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1504948\"]\n<p id=\"fs-id1504948\">exponential decay; The growth factor,[latex]\\,0.825,[\/latex] is between[latex]\\,0\\,[\/latex]and[latex]\\,1.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1461193\">\n<div id=\"fs-id1461195\">\n<p id=\"fs-id1461197\">The population of a herd of deer is represented by the function[latex]\\,A\\left(t\\right)=205{\\left(1.13\\right)}^{t},\\,[\/latex]where[latex]\\,t\\,[\/latex]is given in years. To the nearest whole number, what will the herd population be after[latex]\\,6\\,[\/latex]years?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1658863\">\n<div id=\"fs-id1658865\">\n<p id=\"fs-id1658867\">Find an exponential equation that passes through the points[latex]\\,\\text{(2, 2}\\text{.25)}\\,[\/latex]and[latex]\\,\\left(5,60.75\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1655487\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1655487\"]\n<p id=\"fs-id1655487\">[latex]y=0.25{\\left(3\\right)}^{x}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1394625\">\n<div id=\"fs-id1394627\">\n<p id=\"fs-id1614695\">Determine whether <a class=\"autogenerated-content\" href=\"#Table_04_08_13\">(Figure)<\/a> could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.<\/p>\n\n<table id=\"Table_04_08_13\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)\u201d. Reading the columns as ordered pairs, we have the following values: (1, 3), (2, 0.9), (3, 0.27), and (4, 0.081).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<td>3<\/td>\n<td>0.9<\/td>\n<td>0.27<\/td>\n<td>0.081<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1631360\">\n<div id=\"fs-id1631362\">\n<p id=\"fs-id1631364\">A retirement account is opened with an initial deposit of $8,500 and earns[latex]\\,8.12%\\,[\/latex]interest compounded monthly. What will the account be worth in[latex]\\,20\\,[\/latex]years?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1410744\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1410744\"]\n<p id=\"fs-id1410744\">[latex]$42,888.18[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1676400\">\n<div id=\"fs-id1676402\">\n<p id=\"fs-id1676404\">Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with[latex]\\,7.5%\\,[\/latex]APR, compounded daily, in order to reach her goal in[latex]\\,3\\,[\/latex]years?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1509389\">\n<div id=\"fs-id1509391\">\n<p id=\"fs-id1457722\">Does the equation[latex]\\,y=2.294{e}^{-0.654t}\\,[\/latex]represent continuous growth, continuous decay, or neither? Explain.<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1678302\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1678302\"]\n<p id=\"fs-id1678302\">continuous decay; the growth rate is negative.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1355174\">\n<div id=\"fs-id1355176\">\n<p id=\"fs-id1355178\">Suppose an investment account is opened with an initial deposit of[latex]\\,\\text{\\$10,500}\\,[\/latex]earning[latex]\\,6.25%\\,[\/latex]interest, compounded continuously. How much will the account be worth after[latex]\\,25\\,[\/latex]years?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1407749\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/73c684c9-6dae-4f32-a1a1-5208b5bf59c2\">Graphs of Exponential Functions<\/a><\/h4>\n<div id=\"fs-id1506677\">\n<div id=\"fs-id1506679\">\n<p id=\"fs-id1506681\">Graph the function[latex]\\,f\\left(x\\right)=3.5{\\left(2\\right)}^{x}.\\,[\/latex]State the domain and range and give the <em>y<\/em>-intercept.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1594839\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1594839\"]\n<p id=\"fs-id1594839\">domain: all real numbers; range: all real numbers strictly greater than zero; <em>y<\/em>-intercept: (0, 3.5);<\/p>\n<span id=\"fs-id1430386\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141353\/CNX_PreCalc_Figure_04_08_229.jpg\" alt=\"Graph of f(x)=3.5(2^x)\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1440005\">\n<div id=\"fs-id1440007\">\n<p id=\"fs-id1440009\">Graph the function[latex]\\,f\\left(x\\right)=4{\\left(\\frac{1}{8}\\right)}^{x}\\,[\/latex]and its reflection about the <em>y<\/em>-axis on the same axes, and give the <em>y<\/em>-intercept.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1697407\">\n<div id=\"fs-id1697409\">\n<p id=\"fs-id1697411\">The graph of[latex]\\,f\\left(x\\right)={6.5}^{x}\\,[\/latex]is reflected about the <em>y<\/em>-axis and stretched vertically by a factor of[latex]\\,7.\\,[\/latex]What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1433654\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1433654\"]\n<p id=\"fs-id1433654\">[latex]g\\left(x\\right)=7{\\left(6.5\\right)}^{-x};\\,[\/latex]<em>y<\/em>-intercept:[latex]\\,\\left(0,\\text{ 7}\\right);\\,[\/latex]Domain: all real numbers; Range: all real numbers greater than[latex]\\,0.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405238\">\n<div id=\"fs-id1405240\">\n<p id=\"fs-id1671097\">The graph below shows transformations of the graph of[latex]\\,f\\left(x\\right)={2}^{x}.\\,[\/latex]What is the equation for the transformation?<\/p>\n<span id=\"fs-id1407130\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141405\/CNX_PreCalc_Figure_04_08_231.jpg\" alt=\"Graph of f(x)=2^x\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1586983\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/746b4be7-5dfd-4d01-8293-06ef750e0365\">Logarithmic Functions<\/a><\/h4>\n<div id=\"fs-id1586989\">\n<div id=\"fs-id1350987\">\n<p id=\"fs-id1350989\">Rewrite[latex]\\,{\\mathrm{log}}_{17}\\left(4913\\right)=x\\,[\/latex]as an equivalent exponential equation.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1365015\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1365015\"]\n<p id=\"fs-id1365015\">[latex]{17}^{x}=4913[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1439372\">\n<div id=\"fs-id1439374\">\n<p id=\"fs-id1380764\">Rewrite[latex]\\,\\mathrm{ln}\\left(s\\right)=t\\,[\/latex]as an equivalent exponential equation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1649693\">\n<div id=\"fs-id1649695\">\n<p id=\"fs-id1649697\">Rewrite[latex]\\,{a}^{-\\,\\frac{2}{5}}=b\\,[\/latex]as an equivalent logarithmic equation.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1569161\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1569161\"]\n<p id=\"fs-id1569161\">[latex]{\\mathrm{log}}_{a}b=-\\frac{2}{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1431893\">\n<div id=\"fs-id1431895\">\n<p id=\"fs-id1434949\">Rewrite [latex]\\,{e}^{-3.5}=h\\,[\/latex] as an equivalent logarithmic equation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1631001\">\n<div id=\"fs-id1631003\">\n<p id=\"fs-id1631005\">Solve for x if[latex]\\,\\,\\,{\\mathrm{log}}_{64}\\left(x\\right)=\\frac{1}{3}\\,[\/latex]by converting to exponential form.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1705579\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1705579\"]\n<p id=\"fs-id1705579\">[latex]x={64}^{\\frac{1}{3}}=4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1594722\">\n<div id=\"fs-id1536233\">\n<p id=\"fs-id1536235\">Evaluate[latex]\\,{\\mathrm{log}}_{5}\\left(\\frac{1}{125}\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1658735\">\n<div id=\"fs-id1658737\">\n<p id=\"fs-id1658739\">Evaluate[latex]\\,\\mathrm{log}\\left(\\text{0}\\text{.000001}\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1428405\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1428405\"]\n<p id=\"fs-id1428405\">[latex]\\mathrm{log}\\left(\\text{0}\\text{.000001}\\right)=-6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1517380\">\n<div id=\"fs-id1685591\">\n<p id=\"fs-id1685593\">Evaluate[latex]\\,\\mathrm{log}\\left(4.005\\right)\\,[\/latex]using a calculator. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1536305\">\n<div id=\"fs-id1536307\">\n<p id=\"fs-id1536309\">Evaluate[latex]\\,\\mathrm{ln}\\left({e}^{-0.8648}\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1434281\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1434281\"]\n<p id=\"fs-id1434281\">[latex]\\mathrm{ln}\\left({e}^{-0.8648}\\right)=-0.8648[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405125\">\n<div id=\"fs-id1405127\">\n<p id=\"fs-id1405129\">Evaluate[latex]\\,\\mathrm{ln}\\left(\\sqrt[3]{18}\\right)\\,[\/latex]using a calculator. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1459475\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/44418435-ed46-454a-aba4-cd57f5266654\">Graphs of Logarithmic Functions<\/a><\/h4>\n<div id=\"fs-id1459480\">\n<div id=\"fs-id1459482\">\n<p id=\"fs-id1459484\">Graph the function[latex]\\,g\\left(x\\right)=\\mathrm{log}\\left(7x+21\\right)-4.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1536538\"]Show Solution[\/reveal-answer][hidden-answer a=\"1536538\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141412\/CNX_PreCalc_Figure_04_08_232.jpg\" alt=\"Graph of g(x)=log(7x+21)-4.\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1514032\">\n<div id=\"fs-id1514034\">\n<p id=\"fs-id1514036\">Graph the function[latex]\\,h\\left(x\\right)=2\\mathrm{ln}\\left(9-3x\\right)+1.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1365359\">\n<div id=\"fs-id1588507\">\n<p id=\"fs-id1588509\">State the domain, vertical asymptote, and end behavior of the function[latex]\\,g\\left(x\\right)=\\mathrm{ln}\\left(4x+20\\right)-17.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1701036\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1701036\"]\n<p id=\"fs-id1701036\">Domain:[latex]\\,x&gt;-5;\\,[\/latex]Vertical asymptote:[latex]\\,x=-5;\\,[\/latex]End behavior: as[latex]\\,x\\to -{5}^{+},f\\left(x\\right)\\to -\\infty \\,[\/latex]and as[latex]\\,x\\to \\infty ,f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1807970\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/9d565dd7-7228-45d1-b907-b0e9b7418fb9\">Logarithmic Properties<\/a><\/h4>\n<div id=\"fs-id1865335\">\n<div id=\"fs-id1865338\">\n<p id=\"fs-id1865340\">Rewrite[latex]\\,\\mathrm{ln}\\left(7r\\cdot 11st\\right)\\,[\/latex]in expanded form.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1608999\">\n<div id=\"fs-id1609001\">\n<p id=\"fs-id1609003\">Rewrite[latex]\\,{\\mathrm{log}}_{8}\\left(x\\right)+{\\mathrm{log}}_{8}\\left(5\\right)+{\\mathrm{log}}_{8}\\left(y\\right)+{\\mathrm{log}}_{8}\\left(13\\right)\\,[\/latex]in compact form.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1509129\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1509129\"]\n<p id=\"fs-id1509129\">[latex]{\\text{log}}_{8}\\left(65xy\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1512031\">\n<div id=\"fs-id1512033\">\n<p id=\"fs-id1428552\">Rewrite[latex]\\,{\\mathrm{log}}_{m}\\left(\\frac{67}{83}\\right)\\,[\/latex]in expanded form.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1294263\">\n<div id=\"fs-id1294265\">\n<p id=\"fs-id1600903\">Rewrite[latex]\\,\\mathrm{ln}\\left(z\\right)-\\mathrm{ln}\\left(x\\right)-\\mathrm{ln}\\left(y\\right)\\,[\/latex]in compact form.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1676007\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1676007\"]\n<p id=\"fs-id1676007\">[latex]\\mathrm{ln}\\left(\\frac{z}{xy}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1503660\">\n<div id=\"fs-id1503662\">\n<p id=\"fs-id1503664\">Rewrite[latex]\\,\\mathrm{ln}\\left(\\frac{1}{{x}^{5}}\\right)\\,[\/latex]as a product.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1629102\">\n<div id=\"fs-id1629104\">\n<p id=\"fs-id1707011\">Rewrite[latex]\\,-{\\mathrm{log}}_{y}\\left(\\frac{1}{12}\\right)\\,[\/latex]as a single logarithm.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1371100\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1371100\"]\n<p id=\"fs-id1371100\">[latex]{\\text{log}}_{y}\\left(12\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1433165\">\n<div id=\"fs-id1433167\">\n<p id=\"fs-id1433169\">Use properties of logarithms to expand[latex]\\,\\mathrm{log}\\left(\\frac{{r}^{2}{s}^{11}}{{t}^{14}}\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1584205\">\n<div id=\"fs-id1649231\">\n<p id=\"fs-id1649233\">Use properties of logarithms to expand[latex]\\,\\mathrm{ln}\\left(2b\\sqrt{\\frac{b+1}{b-1}}\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1519261\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1519261\"]\n<p id=\"fs-id1519261\">[latex]\\mathrm{ln}\\left(2\\right)+\\mathrm{ln}\\left(b\\right)+\\frac{\\mathrm{ln}\\left(b+1\\right)-\\mathrm{ln}\\left(b-1\\right)}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1692961\">\n<div id=\"fs-id1443186\">\n<p id=\"fs-id1443188\">Condense the expression[latex]\\,5\\mathrm{ln}\\left(b\\right)+\\mathrm{ln}\\left(c\\right)+\\frac{\\mathrm{ln}\\left(4-a\\right)}{2}\\,[\/latex]to a single logarithm.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1646710\">\n<div id=\"fs-id1646712\">\n<p id=\"fs-id1646714\">Condense the expression[latex]\\,3{\\mathrm{log}}_{7}v+6{\\mathrm{log}}_{7}w-\\frac{{\\mathrm{log}}_{7}u}{3}\\,[\/latex]to a single logarithm.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1419882\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1419882\"]\n<p id=\"fs-id1419882\">[latex]{\\mathrm{log}}_{7}\\left(\\frac{{v}^{3}{w}^{6}}{\\sqrt[3]{u}}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1582444\">\n<div id=\"fs-id1582446\">\n<p id=\"fs-id1582448\">Rewrite[latex]\\,{\\mathrm{log}}_{3}\\left(12.75\\right)\\,[\/latex]to base[latex]\\,e.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1429357\">\n<div id=\"fs-id1429359\">\n<p id=\"fs-id1429361\">Rewrite[latex]\\,{5}^{12x-17}=125\\,[\/latex]as a logarithm. Then apply the change of base formula to solve for[latex]\\,x\\,[\/latex]using the common log. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1352011\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1352011\"]\n<p id=\"fs-id1352011\">[latex]x=\\frac{\\frac{\\mathrm{log}\\left(125\\right)}{\\mathrm{log}\\left(5\\right)}+17}{12}=\\frac{5}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1376798\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/c1f8641f-2121-4457-adc2-ef58f23500ce\">Exponential and Logarithmic Equations<\/a><\/h4>\n<div id=\"fs-id1376803\">\n<div id=\"fs-id1376805\">\n<p id=\"fs-id1376808\">Solve[latex]\\,{216}^{3x}\\cdot {216}^{x}={36}^{3x+2}\\,[\/latex]by rewriting each side with a common base.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1421086\">\n<div id=\"fs-id1421088\">\n<p id=\"fs-id1691321\">Solve[latex]\\,\\frac{125}{{\\left(\\frac{1}{625}\\right)}^{-x-3}}={5}^{3}\\,[\/latex]by rewriting each side with a common base.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1376667\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1376667\"]\n<p id=\"fs-id1376667\">[latex]x=-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1431560\">\n<div id=\"fs-id1431562\">\n<p id=\"fs-id1431564\">Use logarithms to find the exact solution for[latex]\\,7\\cdot {17}^{-9x}-7=49.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1563478\">\n<div id=\"fs-id1434653\">\n<p id=\"fs-id1434655\">Use logarithms to find the exact solution for[latex]\\,3{e}^{6n-2}+1=-60.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1582355\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1582355\"]\n<p id=\"fs-id1582355\">no solution<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1582361\">\n<div id=\"fs-id1424610\">\n<p id=\"fs-id1424612\">Find the exact solution for[latex]\\,5{e}^{3x}-4=6\\,[\/latex]. If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1608922\">\n<div id=\"fs-id1518514\">\n<p id=\"fs-id1518516\">Find the exact solution for[latex]\\,2{e}^{5x-2}-9=-56.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1671694\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1671694\"]\n<p id=\"fs-id1671694\">no solution<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1671700\">\n<div id=\"fs-id1602116\">\n<p id=\"fs-id1602118\">Find the exact solution for[latex]\\,{5}^{2x-3}={7}^{x+1}.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1570009\">\n<div id=\"fs-id1570011\">\n<p id=\"fs-id1570013\">Find the exact solution for[latex]\\,{e}^{2x}-{e}^{x}-110=0.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1403522\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1403522\"]\n<p id=\"fs-id1403522\">[latex]x=\\mathrm{ln}\\left(11\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1698182\">\n<div id=\"fs-id1698185\">\n<p id=\"fs-id1698187\">Use the definition of a logarithm to solve.[latex]\\,-5{\\mathrm{log}}_{7}\\left(10n\\right)=5.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1408770\">\n<div id=\"fs-id1408772\">\n<p id=\"fs-id1408774\">47. Use the definition of a logarithm to find the exact solution for[latex]\\,9+6\\mathrm{ln}\\left(a+3\\right)=33.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1609098\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1609098\"]\n<p id=\"fs-id1609098\">[latex]a={e}^{4}-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1446674\">\n<div id=\"fs-id1446676\">\n<p id=\"fs-id1446678\">Use the one-to-one property of logarithms to find an exact solution for[latex]\\,{\\mathrm{log}}_{8}\\left(7\\right)+{\\mathrm{log}}_{8}\\left(-4x\\right)={\\mathrm{log}}_{8}\\left(5\\right).\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405530\">\n<div id=\"fs-id1440096\">\n<p id=\"fs-id1440098\">Use the one-to-one property of logarithms to find an exact solution for[latex]\\,\\mathrm{ln}\\left(5\\right)+\\mathrm{ln}\\left(5{x}^{2}-5\\right)=\\mathrm{ln}\\left(56\\right).\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1408805\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1408805\"]\n<p id=\"fs-id1408805\">[latex]x=\u00b1\\frac{9}{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1395438\">\n<div id=\"fs-id1395440\">\n<p id=\"fs-id1409989\">The formula for measuring sound intensity in decibels[latex]\\,D\\,[\/latex]is defined by the equation[latex]\\,D=10\\mathrm{log}\\left(\\frac{I}{{I}_{0}}\\right),[\/latex] where[latex]\\,I\\,[\/latex]is the intensity of the sound in watts per square meter and[latex]\\,{I}_{0}={10}^{-12}\\,[\/latex]is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of[latex]\\,6.3\\cdot {10}^{-3}\\,[\/latex]watts per square meter?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1376365\">\n<div id=\"fs-id1520052\">\n<p id=\"fs-id1520055\">The population of a city is modeled by the equation[latex]\\,P\\left(t\\right)=256,114{e}^{0.25t}\\,[\/latex]where[latex]\\,t\\,[\/latex]is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1648231\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1648231\"]\n<p id=\"fs-id1648231\">about[latex]\\,5.45\\,[\/latex]years<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1624824\">\n<div id=\"fs-id1624827\">\n<p id=\"fs-id1624829\">Find the inverse function[latex]\\,{f}^{-1}\\,[\/latex]for the exponential function[latex]\\,f\\left(x\\right)=2\\cdot {e}^{x+1}-5.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1520404\">\n<div id=\"fs-id1520406\">\n<p id=\"fs-id1520408\">Find the inverse function[latex]\\,{f}^{-1}\\,[\/latex]for the logarithmic function[latex]\\,f\\left(x\\right)=0.25\\cdot {\\mathrm{log}}_{2}\\left({x}^{3}+1\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1563604\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1563604\"]\n<p id=\"fs-id1563604\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt[3]{{2}^{4x}-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1563711\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/feda96a1-a0f3-41ce-9d42-43eef361a909\">Exponential and Logarithmic Models<\/a><\/h4>\n<p id=\"fs-id1563717\">For the following exercises, use this scenario: A doctor prescribes[latex]\\,300\\,[\/latex]milligrams of a therapeutic drug that decays by about[latex]\\,17%\\,[\/latex]each hour.<\/p>\n\n<div id=\"fs-id1452308\">\n<div id=\"fs-id1393659\">\n<p id=\"fs-id1393661\">To the nearest minute, what is the half-life of the drug?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1393666\">\n<div id=\"fs-id1393668\">\n<p id=\"fs-id1393670\">Write an exponential model representing the amount of the drug remaining in the patient\u2019s system after[latex]\\,t\\,[\/latex]hours. Then use the formula to find the amount of the drug that would remain in the patient\u2019s system after[latex]\\,24\\,[\/latex]hours. Round to the nearest hundredth of a gram.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1671547\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1671547\"]\n<p id=\"fs-id1671547\">[latex]f\\left(t\\right)=300{\\left(0.83\\right)}^{t};f\\left(24\\right)\\approx 3.43\\text{\u200a}\\text{\u200a}g[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1647206\">For the following exercises, use this scenario: A soup with an internal temperature of[latex]\\,\\text{350\u00b0}\\,[\/latex]Fahrenheit was taken off the stove to cool in a[latex]\\,\\text{71\u00b0F}\\,[\/latex]room. After fifteen minutes, the internal temperature of the soup was[latex]\\,\\text{175\u00b0F}\\text{.}[\/latex]<\/p>\n\n<div id=\"fs-id1410718\">\n<div id=\"fs-id1410720\">\n<p id=\"fs-id1410722\">Use Newton\u2019s Law of Cooling to write a formula that models this situation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1410729\">\n<div id=\"fs-id1410731\">\n<p id=\"fs-id1422416\">How many minutes will it take the soup to cool to[latex]\\,\\text{85\u00b0F?}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1409376\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1409376\"]\n<p id=\"fs-id1409376\">about[latex]\\,45\\,[\/latex]minutes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1354791\">For the following exercises, use this scenario: The equation[latex]\\,N\\left(t\\right)=\\frac{1200}{1+199{e}^{-0.625t}}\\,[\/latex]models the number of people in a school who have heard a rumor after[latex]\\,t\\,[\/latex]days.<\/p>\n\n<div id=\"fs-id1406672\">\n<div id=\"fs-id1406674\">\n<p id=\"fs-id1406676\">How many people started the rumor?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1406889\">\n<div id=\"fs-id1406892\">\n<p id=\"fs-id1406894\">To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1406900\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1406900\"]\n<p id=\"fs-id1406900\">about[latex]\\,8.5\\,[\/latex]days<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1512099\">\n<div id=\"fs-id1512102\">\n<p id=\"fs-id1512104\">What is the carrying capacity?<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1512109\">For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.<\/p>\n\n<div id=\"fs-id1652654\">\n<div id=\"fs-id1652656\">\n<table id=\"fs-id1652658\" class=\"unnumbered\" summary=\"Two columns and eleven rpws. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 3.05), (2, 4.42), (3, 6.4), (4, 9.28), (5, 13.46), (6, 19.52), (7, 28.3), (8, 41.01), (9, 59.5), and (10, 86.28).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>3.05<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>4.42<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>6.4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>9.28<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>13.46<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>19.52<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>28.3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>41.04<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>59.5<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>86.28<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1403487\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1403487\"]\n<p id=\"fs-id1403487\">exponential<\/p>\n<span id=\"fs-id1406597\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141414\/CNX_PreCalc_Figure_04_08_234.jpg\" alt=\"Graph of the table\u2019s values.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1423620\">\n<div id=\"fs-id1423623\">\n<table id=\"fs-id1423625\" class=\"unnumbered\" summary=\"Two columns and twelve rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (0.5, 18.05), (1, 17), (3, 15.33), (5, 14.55), (7, 14.04), (10, 13.5), (12, 13.22), (13, 13.1), (15, 12.88), (17, 12.69), and (20, 12.45).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>0.5<\/td>\n<td>18.05<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>17<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>15.33<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>14.55<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>14.04<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>13.5<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>13.22<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>13.1<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>12.88<\/td>\n<\/tr>\n<tr>\n<td>17<\/td>\n<td>12.69<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>12.45<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1614793\">\n<div id=\"fs-id1614795\">\n<p id=\"fs-id1614797\">Find a formula for an exponential equation that goes through the points[latex]\\,\\left(-2,100\\right)\\,[\/latex]and[latex]\\,\\left(0,4\\right).\\,[\/latex]Then express the formula as an equivalent equation with base <em>e.<\/em><\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1657094\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1657094\"]\n<p id=\"fs-id1657094\">[latex]y=4{\\left(0.2\\right)}^{x};\\,[\/latex][latex]y=4{e}^{\\text{-1}\\text{.609438}x}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1658476\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/aa3a6479-56b9-4e74-9b72-5817cfe5bf5a\">Fitting Exponential Models to Data<\/a><\/h4>\n<div id=\"fs-id1658482\">\n<div id=\"fs-id1699077\">\n<p id=\"fs-id1699079\">What is the carrying capacity for a population modeled by the logistic equation[latex]\\,P\\left(t\\right)=\\frac{250,000}{1\\,\\,+\\,\\,499{e}^{-0.45t}}?\\,[\/latex]What is the initial population for the model?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1446741\">\n<div id=\"fs-id1446744\">\n<p id=\"fs-id1424342\">The population of a culture of bacteria is modeled by the logistic equation[latex]\\,P\\left(t\\right)=\\frac{14,250}{1\\,\\,+\\,\\,29{e}^{-0.62t}},[\/latex] where[latex]\\,t\\,[\/latex]is in days. To the nearest tenth, how many days will it take the culture to reach[latex]\\,75%\\,[\/latex]of its carrying capacity?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1696243\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1696243\"]\n<p id=\"fs-id1696243\">about[latex]\\,7.2\\,[\/latex]days<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1562633\">For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.<\/p>\n\n<div id=\"fs-id1597899\">\n<div id=\"fs-id1597901\">\n<table id=\"fs-id1597902\" class=\"unnumbered\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 409.4), (2, 260.7), (3, 170.4), (4, 110.6), (5, 74), (6, 44.7), (7, 32.4), (8, 19.5), (9, 12.7), and (10, 8.1).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>409.4<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>260.7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>170.4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>110.6<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>74<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>44.7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>32.4<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>19.5<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>12.7<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>8.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1761662\">\n<div id=\"fs-id1761665\">\n<table id=\"fs-id1761667\" class=\"unnumbered\" summary=\"Two rows and twelve columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)\u201d. Reading the columns as ordered pairs, we have the following values: (0.15, 36.21), (0.25, 28.88), (0.5, 24.39), (0.75, 18.28), (1, 16.5), (1.5, 12.99), (2, 9.91), (2.25, 8.57), (2.75, 7.23), (3, 5.99), and (3.5, 4.81).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>0.15<\/td>\n<td>36.21<\/td>\n<\/tr>\n<tr>\n<td>0.25<\/td>\n<td>28.88<\/td>\n<\/tr>\n<tr>\n<td>0.5<\/td>\n<td>24.39<\/td>\n<\/tr>\n<tr>\n<td>0.75<\/td>\n<td>18.28<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>16.5<\/td>\n<\/tr>\n<tr>\n<td>1.5<\/td>\n<td>12.99<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>9.91<\/td>\n<\/tr>\n<tr>\n<td>2.25<\/td>\n<td>8.57<\/td>\n<\/tr>\n<tr>\n<td>2.75<\/td>\n<td>7.23<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>5.99<\/td>\n<\/tr>\n<tr>\n<td>3.5<\/td>\n<td>4.81<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1705002\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1705002\"]\n<p id=\"fs-id1705002\">logarithmic;[latex]\\,y=16.68718-9.71860\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<span id=\"fs-id1421423\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141434\/CNX_PreCalc_Figure_04_08_237.jpg\" alt=\"Graph of the table\u2019s values.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1408475\">\n<div id=\"fs-id1408477\">\n<table id=\"fs-id1408479\" class=\"unnumbered\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (0, 9), (2, 22.6), (4, 44.2), (5, 62.1), (7, 96.9), (8, 113.4), (10, 133.4), (11, 137.6), (15, 148.4), and (17, 149.3).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>22.6<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>44.2<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>62.1<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>96.9<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>113.4<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>133.4<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>137.6<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>148.4<\/td>\n<\/tr>\n<tr>\n<td>17<\/td>\n<td>149.3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1513725\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<div id=\"fs-id1513728\">\n<div id=\"fs-id1513730\">\n<p id=\"fs-id1513732\">The population of a pod of bottlenose dolphins is modeled by the function[latex]\\,A\\left(t\\right)=8{\\left(1.17\\right)}^{t},[\/latex] where[latex]\\,t\\,[\/latex]is given in years. To the nearest whole number, what will the pod population be after[latex]\\,3\\,[\/latex]years?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1597336\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1597336\"]\n<p id=\"fs-id1597336\">About[latex]\\,13\\,[\/latex]dolphins.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1664103\">\n<div id=\"fs-id1664105\">\n<p id=\"fs-id1664107\">Find an exponential equation that passes through the points[latex]\\,\\text{(0, 4)}\\,[\/latex]and[latex]\\,\\text{(2, 9)}\\text{.}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1530968\">\n<div id=\"fs-id1530970\">\n<p id=\"fs-id1530972\">Drew wants to save $2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with[latex]\\,6.25%\\,[\/latex]APR, compounding daily, in order to reach his goal in[latex]\\,4\\,[\/latex]years?<\/p>\n\n<\/div>\n<div id=\"fs-id723927\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id723927\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id723927\"]\n<p id=\"fs-id723929\">[latex]$1,947[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1457825\">\n<div id=\"fs-id1457827\">\n<p id=\"fs-id1457829\">An investment account was opened with an initial deposit of $9,600 and earns[latex]\\,7.4%\\,[\/latex]interest, compounded continuously. How much will the account be worth after[latex]\\,15\\,[\/latex]years?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1375232\">\n<div id=\"fs-id1375234\">\n<p id=\"fs-id1375236\">Graph the function[latex]\\,f\\left(x\\right)=5{\\left(0.5\\right)}^{-x}\\,[\/latex]and its reflection across the <em>y<\/em>-axis on the same axes, and give the <em>y<\/em>-intercept.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1588458\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1588458\"]\n<p id=\"fs-id1588458\"><em>y<\/em>-intercept:[latex]\\,\\left(0,\\text{ 5}\\right)[\/latex]<\/p>\n<span id=\"fs-id1357638\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141444\/CNX_PreCalc_Figure_04_08_239.jpg\" alt=\"Graph of f(-x)=5(0.5)^-x in blue and f(x)=5(0.5)^x in orange.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1691088\">\n<div id=\"fs-id1691090\">\n<p id=\"fs-id1691092\">The graph shows transformations of the graph of[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.\\,[\/latex]What is the equation for the transformation?<\/p>\n<span id=\"fs-id1586380\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141451\/CNX_PreCalc_Figure_04_08_240.jpg\" alt=\"Graph of f(x)= (1\/2)^x.\"><\/span>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1407487\">\n<div id=\"fs-id1407489\">\n<p id=\"fs-id1407491\">Rewrite[latex]\\,{\\mathrm{log}}_{8.5}\\left(614.125\\right)=a\\,[\/latex]as an equivalent exponential equation.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1653046\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1653046\"]\n<p id=\"fs-id1653046\">[latex]{8.5}^{a}=614.125[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1487536\">\n<div id=\"fs-id1487538\">\n<p id=\"fs-id1338247\">Rewrite[latex]\\,{e}^{\\frac{1}{2}}=m\\,[\/latex]as an equivalent logarithmic equation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1393960\">\n<div id=\"fs-id1393962\">\n<p id=\"fs-id1393964\">Solve for[latex]\\,x\\,[\/latex]by converting the logarithmic equation[latex]\\,lo{g}_{\\frac{1}{7}}\\left(x\\right)=2\\,[\/latex]to exponential form.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1361043\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1361043\"]\n<p id=\"fs-id1361043\">[latex]x={\\left(\\frac{1}{7}\\right)}^{2}=\\frac{1}{49}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1630279\">\n<div id=\"fs-id1630281\">\n<p id=\"fs-id1630283\">Evaluate[latex]\\,\\mathrm{log}\\left(\\text{10,000,000}\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1430674\">\n<div id=\"fs-id1423430\">\n<p id=\"fs-id1423432\">Evaluate[latex]\\,\\mathrm{ln}\\left(0.716\\right)\\,[\/latex]using a calculator. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1460620\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1460620\"]\n<p id=\"fs-id1460620\">[latex]\\mathrm{ln}\\left(0.716\\right)\\approx -0.334[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405885\">\n<div id=\"fs-id1405888\">\n<p id=\"fs-id1405890\">Graph the function[latex]\\,g\\left(x\\right)=\\mathrm{log}\\left(12-6x\\right)+3.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1651954\">\n<div id=\"fs-id1651956\">\n<p id=\"fs-id1651958\">State the domain, vertical asymptote, and end behavior of the function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{5}\\left(39-13x\\right)+7.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1678313\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1678313\"]\n<p id=\"fs-id1678313\">Domain:[latex]\\,x&lt;3;\\,[\/latex]Vertical asymptote:[latex]\\,x=3;\\,[\/latex]End behavior:[latex]\\,x\\to {3}^{-},f\\left(x\\right)\\to -\\infty \\,[\/latex]and[latex]\\,x\\to -\\infty ,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1423972\">\n<div id=\"fs-id1423974\">\n<p id=\"fs-id1423976\">Rewrite[latex]\\,\\mathrm{log}\\left(17a\\cdot 2b\\right)\\,[\/latex]as a sum.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1508077\">\n<div id=\"fs-id1430560\">\n<p id=\"fs-id1430562\">Rewrite[latex]\\,{\\mathrm{log}}_{t}\\left(96\\right)-{\\mathrm{log}}_{t}\\left(8\\right)\\,[\/latex]in compact form.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1406644\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1406644\"]\n<p id=\"fs-id1406644\">[latex]{\\mathrm{log}}_{t}\\left(12\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405262\">\n<div id=\"fs-id1405264\">\n<p id=\"fs-id1405267\">Rewrite[latex]\\,{\\mathrm{log}}_{8}\\left({a}^{\\frac{1}{b}}\\right)\\,[\/latex]as a product.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1690738\">\n<div id=\"fs-id1690740\">\n<p id=\"fs-id1690742\">Use properties of logarithm to expand[latex]\\,\\mathrm{ln}\\left({y}^{3}{z}^{2}\\cdot \\sqrt[3]{x-4}\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1439808\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1439808\"]\n<p id=\"fs-id1439808\">[latex]3\\,\\,\\mathrm{ln}\\left(y\\right)+2\\mathrm{ln}\\left(z\\right)+\\frac{\\mathrm{ln}\\left(x-4\\right)}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1660668\">\n<div id=\"fs-id1660670\">\n<p id=\"fs-id1660672\">Condense the expression[latex]\\,4\\mathrm{ln}\\left(c\\right)+\\mathrm{ln}\\left(d\\right)+\\frac{\\mathrm{ln}\\left(a\\right)}{3}+\\frac{\\mathrm{ln}\\left(b+3\\right)}{3}\\,[\/latex]to a single logarithm.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1417371\">\n<div id=\"fs-id1417373\">\n<p id=\"fs-id1528728\">Rewrite[latex]\\,{16}^{3x-5}=1000\\,[\/latex]as a logarithm. Then apply the change of base formula to solve for[latex]\\,x\\,[\/latex]using the natural log. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1647240\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1647240\"]\n<p id=\"fs-id1647240\">[latex]x=\\frac{\\frac{\\mathrm{ln}\\left(1000\\right)}{\\mathrm{ln}\\left(16\\right)}+5}{3}\\approx 2.497[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1395132\">\n<div id=\"fs-id1395134\">\n<p id=\"fs-id1395136\">Solve[latex]\\,{\\left(\\frac{1}{81}\\right)}^{x}\\cdot \\frac{1}{243}={\\left(\\frac{1}{9}\\right)}^{-3x-1}\\,[\/latex]by rewriting each side with a common base.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1646539\">\n<div id=\"fs-id1646541\">\n<p id=\"fs-id1646543\">Use logarithms to find the exact solution for[latex]\\,-9{e}^{10a-8}-5=-41[\/latex]. If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1700907\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1700907\"]\n<p id=\"fs-id1700907\">[latex]a=\\frac{\\mathrm{ln}\\left(4\\right)+8}{10}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1365542\">\n<div id=\"fs-id1365544\">\n<p id=\"fs-id1365546\">Find the exact solution for[latex]\\,10{e}^{4x+2}+5=56.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1522787\">\n<div id=\"fs-id1522789\">\n<p id=\"fs-id1522792\">Find the exact solution for[latex]\\,-5{e}^{-4x-1}-4=64.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1402901\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1402901\"]\n<p id=\"fs-id1402901\">no solution<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1402906\">\n<div id=\"fs-id1402909\">\n<p id=\"fs-id1402911\">Find the exact solution for[latex]\\,{2}^{x-3}={6}^{2x-1}.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1407851\">\n<div id=\"fs-id1407853\">\n<p id=\"fs-id1407855\">Find the exact solution for[latex]\\,{e}^{2x}-{e}^{x}-72=0.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1434398\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1434398\"]\n<p id=\"fs-id1434398\">[latex]x=\\mathrm{ln}\\left(9\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1435523\">\n<div id=\"fs-id1435525\">\n<p id=\"fs-id1435527\">Use the definition of a logarithm to find the exact solution for[latex]\\,4\\mathrm{log}\\left(2n\\right)-7=-11[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1589316\">\n<div id=\"fs-id1589318\">\n<p id=\"fs-id1589320\">Use the one-to-one property of logarithms to find an exact solution for[latex]\\,\\mathrm{log}\\left(4{x}^{2}-10\\right)+\\mathrm{log}\\left(3\\right)=\\mathrm{log}\\left(51\\right)\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1435205\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1435205\"]\n<p id=\"fs-id1435205\">[latex]x=\u00b1\\frac{3\\sqrt{3}}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1408912\">\n<div id=\"fs-id1408914\">\n<p id=\"fs-id1522030\">The formula for measuring sound intensity in decibels[latex]\\,D\\,[\/latex]is defined by the equation[latex]\\,D=10\\mathrm{log}\\left(\\frac{I}{{I}_{0}}\\right),[\/latex]where[latex]\\,I\\,[\/latex]is the intensity of the sound in watts per square meter and[latex]\\,{I}_{0}={10}^{-12}\\,[\/latex]is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of[latex]\\,4.7\\cdot {10}^{-1}\\,[\/latex]watts per square meter?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1457675\">\n<div id=\"fs-id1457677\">\n<p id=\"fs-id1457680\">A radiation safety officer is working with[latex]\\,112\\,[\/latex]grams of a radioactive substance. After[latex]\\,17\\,[\/latex]days, the sample has decayed to[latex]\\,80\\,[\/latex]grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1654858\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1654858\"][latex]f\\left(t\\right)=112{e}^{-.019792t};[\/latex] half-life: about[latex]\\,35\\,[\/latex] days[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1432306\">\n<div id=\"fs-id1432309\">\n<p id=\"fs-id1432311\">Write the formula found in the previous exercise as an equivalent equation with base[latex]\\,e.\\,[\/latex]Express the exponent to five significant digits.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1405159\">\n<div id=\"fs-id1405161\">\n<p id=\"fs-id1405163\">A bottle of soda with a temperature of[latex]\\,\\text{71\u00b0}\\,[\/latex]Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of[latex]\\,\\text{35\u00b0 F}\\text{.}\\,[\/latex]After ten minutes, the internal temperature of the soda was[latex]\\,\\text{63\u00b0 F}\\text{.}\\,[\/latex]Use Newton\u2019s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1700652\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1700652\"]\n<p id=\"fs-id1700652\">[latex]T\\left(t\\right)=36{e}^{-0.025131t}+35;T\\left(60\\right)\\approx {43}^{\\text{o}}\\text{F}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1487596\">\n<div id=\"fs-id1504591\">\n<p id=\"fs-id1504593\">The population of a wildlife habitat is modeled by the equation[latex]\\,P\\left(t\\right)=\\frac{360}{1+6.2{e}^{-0.35t}},[\/latex] where[latex]\\,t\\,[\/latex]is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1517136\">\n<div id=\"fs-id1517138\">\n\nEnter the data from <a class=\"autogenerated-content\" href=\"#Table_04_08_14\">(Figure)<\/a> into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.\n<table id=\"Table_04_08_14\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 3), (2, 8.55), (3, 11.79), (4, 14.09), (5, 15.88), (6, 17.33), (7, 18.57), (8, 19.64), (9, 20.58), and (10, 21.42).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>8.55<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>11.79<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>14.09<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>15.88<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>17.33<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>18.57<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>19.64<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>20.58<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>21.42<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1665023\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1665023\"]\n<p id=\"fs-id1665023\">logarithmic<\/p>\n<span id=\"fs-id1665029\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141505\/CNX_PreCalc_Figure_04_08_242.jpg\" alt=\"Graph of the table\u2019s values.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1665041\">\n<div id=\"fs-id1665043\">\n<p id=\"fs-id1678197\">The population of a lake of fish is modeled by the logistic equation[latex]\\,P\\left(t\\right)=\\frac{16,120}{1+25{e}^{-0.75t}},[\/latex] where[latex]\\,t\\,[\/latex]is time in years. To the nearest hundredth, how many years will it take the lake to reach[latex]\\,80%\\,[\/latex]of its carrying capacity?<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1705179\">For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.<\/p>\n\n<div id=\"fs-id1705185\">\n<div id=\"fs-id1600290\">\n<table id=\"fs-id1600292\" class=\"unnumbered\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 20), (2, 21.6), (3, 29.2), (4, 36.4), (5, 46.6), (6, 55.7), (7, 72.6), (8, 87.1), (9, 107.2), and (10, 138.1).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>20<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>21.6<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>29.2<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>36.4<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>46.6<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>55.7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>72.6<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>87.1<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>107.2<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>138.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1465082\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1465082\"]\n<p id=\"fs-id1465082\">exponential;[latex]\\,y=15.10062{\\left(1.24621\\right)}^{x}[\/latex]<\/p>\n<span id=\"fs-id1512245\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141508\/CNX_PreCalc_Figure_04_08_243.jpg\" alt=\"Graph of the table\u2019s values.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1300030\">\n<div id=\"fs-id1300032\">\n<table id=\"fs-id1300035\" class=\"unnumbered\" summary=\"Two columns and twelve rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (3, 13.98), (4, 17.84), (5, 20.01), (6, 22.7), (7, 24.1), (8, 26.15), (9, 27.37), (10, 28.38), (11, 29.97), (12, 31.07), and (13, 31.43).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>13.98<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>17.84<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>20.01<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>22.7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>24.1<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>26.15<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>27.37<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>28.38<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>29.97<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>31.07<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>31.43<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1676036\">\n<div id=\"fs-id1676038\">\n<table id=\"fs-id1676040\" class=\"unnumbered\" summary=\"Two columns and twelve rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (0, 2.2), (0.5, 2.9), (1, 3.9), (1.5,4.8), (2, 6.4), (3, 9.3), (4, 12.3), (5, 15), (6, 16.2), (7, 17.3), and (8, 17.9).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>2.2<\/td>\n<\/tr>\n<tr>\n<td>0.5<\/td>\n<td>2.9<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>3.9<\/td>\n<\/tr>\n<tr>\n<td>1.5<\/td>\n<td>4.8<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>6.4<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>9.3<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>12.3<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>15<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>16.2<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>17.3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>17.9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1503926\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1503926\"]\n<p id=\"fs-id1503926\">logistic;[latex]\\,y=\\frac{18.41659}{1+7.54644{e}^{-0.68375x}}[\/latex]<\/p>\n<span id=\"fs-id1406292\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141524\/CNX_PreCalc_Figure_04_08_245.jpg\" alt=\"Graph of the table\u2019s values.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Build an exponential model from data.<\/li>\n<li>Build a logarithmic model from data.<\/li>\n<li>Build a logistic model from data.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1677675\">In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called <em>regression analysis<\/em> to find a curve that models data collected from real-world observations. With <span class=\"no-emphasis\">regression analysis<\/span>, we don\u2019t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events.<\/p>\n<p id=\"fs-id1356390\">Do not be confused by the word <em>model<\/em>. In mathematics, we often use the terms <em>function<\/em>, <em>equation<\/em>, and <em>model<\/em> interchangeably, even though they each have their own formal definition. The term <em>model<\/em> is typically used to indicate that the equation or function approximates a real-world situation.<\/p>\n<p id=\"fs-id969059\">We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work we\u2019ve done so far, and then explore the ways regression is used to model real-world phenomena.<\/p>\n<div id=\"fs-id1527075\" class=\"bc-section section\">\n<h3>Building an Exponential Model from Data<\/h3>\n<p id=\"fs-id1637870\">As we\u2019ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that\u2019s not the whole story. It\u2019s the <em>way<\/em> data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let\u2019s review exponential growth and decay.<\/p>\n<p id=\"fs-id1365833\">Recall that exponential functions have the form[latex]\\,y=a{b}^{x}\\,[\/latex]or[latex]\\,y={A}_{0}{e}^{kx}.\\,[\/latex]When performing regression analysis, we use the form most commonly used on graphing utilities,[latex]\\,y=a{b}^{x}.\\,[\/latex]Take a moment to reflect on the characteristics we\u2019ve already learned about the exponential function[latex]\\,y=a{b}^{x}\\,[\/latex](assume[latex]\\,a>0[\/latex]):<\/p>\n<ul id=\"fs-id1694862\">\n<li>[latex]b\\,[\/latex]must be greater than zero and not equal to one.<\/li>\n<li>The initial value of the model is[latex]\\,y=a.[\/latex]\n<ul id=\"fs-id1305009\">\n<li>If[latex]\\,b>1,[\/latex] the function models exponential growth. As[latex]\\,x\\,[\/latex]increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound.<\/li>\n<li>If[latex]\\,0<b<1,[\/latex] the function models <span class=\"no-emphasis\">exponential decay<\/span>. As[latex]\\,x\\,[\/latex]increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the <em>x<\/em>-axis. In other words, the outputs never become equal to or less than zero.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p id=\"fs-id1373960\">As part of the results, your calculator will display a number known as the <em>correlation coefficient<\/em>, labeled by the variable[latex]\\,r,[\/latex] or[latex]\\,{r}^{2}.\\,[\/latex](You may have to change the calculator\u2019s settings for these to be shown.) The values are an indication of the \u201cgoodness of fit\u201d of the regression equation to the data. We more commonly use the value of[latex]\\,{r}^{2}\\,[\/latex]instead of[latex]\\,r,[\/latex] but the closer either value is to 1, the better the regression equation approximates the data.<\/p>\n<div id=\"fs-id1424609\" class=\"textbox key-takeaways\">\n<h3>Exponential Regression<\/h3>\n<p id=\"fs-id1532328\"><em>Exponential regression<\/em> is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command \u201cExpReg\u201d on a graphing utility to fit an exponential function to a set of data points. This returns an equation of the form,[latex]y=a{b}^{x}[\/latex]<\/p>\n<p id=\"fs-id1585350\">Note that:<\/p>\n<ul id=\"fs-id1534713\">\n<li>[latex]b\\,[\/latex]must be non-negative.<\/li>\n<li>when[latex]\\,b>1,[\/latex] we have an exponential growth model.<\/li>\n<li>when[latex]\\,0<b<1,[\/latex] we have an exponential decay model.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1375157\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1588505\"><strong>Given a set of data, perform exponential regression using a graphing utility.<\/strong><\/p>\n<ol id=\"fs-id836076\" type=\"1\">\n<li>Use the STAT then EDIT menu to enter given data.\n<ol id=\"fs-id1293089\" type=\"a\">\n<li>Clear any existing data from the lists.<\/li>\n<li>List the input values in the L1 column.<\/li>\n<li>List the output values in the L2 column.<\/li>\n<\/ol>\n<\/li>\n<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\n<ol id=\"fs-id900017\" type=\"a\">\n<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\n<li>Verify the data follow an exponential pattern.<\/li>\n<\/ol>\n<\/li>\n<li>Find the equation that models the data.\n<ol id=\"fs-id1677252\" type=\"a\">\n<li>Select \u201cExpReg\u201d from the STAT then CALC menu.<\/li>\n<li>Use the values returned for <em>a<\/em> and <em>b<\/em> to record the model,[latex]\\,y=a{b}^{x}.[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_08_01\" class=\"textbox examples\">\n<div id=\"fs-id1647375\">\n<div id=\"fs-id1338736\">\n<h3>Using Exponential Regression to Fit a Model to Data<\/h3>\n<p id=\"fs-id1424549\">In 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from 2,871 crashes were used to measure the association of a person\u2019s blood alcohol level (BAC) with the risk of being in an accident. <a class=\"autogenerated-content\" href=\"#Table_04_08_01\">(Figure)<\/a> shows results from the study<a class=\"footnote\" title=\"\u2022Source: Indiana University Center for Studies of Law in Action, 2007\" id=\"return-footnote-112-1\" href=\"#footnote-112-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> . The <em>relative risk<\/em> is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of 0.09 is 3.54 times as likely to crash as a person who has not been drinking alcohol.<\/p>\n<table id=\"Table_04_08_01\" summary=\"Two rows and thirteen columns. The first row is labeled, \u201cBAC\u201d, and the second row is labeled, \u201cRelative Risk of Crashing\u201d. Reading the columns as ordered pairs, we have the following values: (0, 1), (0.01, 1.03), (0.03, 1.06), (0.05, 1.38), (0.07, 2.09), (0.09, 3.54), (0.11, 6.41), (0.13, 12.6), (0.15, 22.1), (0.17, 39.05), (0.19, 65.32), and (0.21, 4.394).\">\n<tbody>\n<tr>\n<td><strong>BAC<\/strong><\/td>\n<td>0<\/td>\n<td>0.01<\/td>\n<td>0.03<\/td>\n<td>0.05<\/td>\n<td>0.07<\/td>\n<td>0.09<\/td>\n<\/tr>\n<tr>\n<td><strong>Relative Risk of Crashing<\/strong><\/td>\n<td>1<\/td>\n<td>1.03<\/td>\n<td>1.06<\/td>\n<td>1.38<\/td>\n<td>2.09<\/td>\n<td>3.54<\/td>\n<\/tr>\n<tr>\n<td><strong>BAC<\/strong><\/td>\n<td>0.11<\/td>\n<td>0.13<\/td>\n<td>0.15<\/td>\n<td>0.17<\/td>\n<td>0.19<\/td>\n<td>0.21<\/td>\n<\/tr>\n<tr>\n<td><strong>Relative Risk of Crashing<\/strong><\/td>\n<td>6.41<\/td>\n<td>12.6<\/td>\n<td>22.1<\/td>\n<td>39.05<\/td>\n<td>65.32<\/td>\n<td>99.78<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1326068\" type=\"a\">\n<li>Let[latex]\\,x\\,[\/latex]represent the BAC level, and let[latex]\\,y\\,[\/latex]represent the corresponding relative risk. Use exponential regression to fit a model to these data.<\/li>\n<li>After 6 drinks, a person weighing 160 pounds will have a BAC of about[latex]\\,0.16.\\,[\/latex]How many times more likely is a person with this weight to crash if they drive after having a 6-pack of beer? Round to the nearest hundredth.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1588497\" type=\"a\">\n<li>Using the STAT then EDIT menu on a graphing utility, list the BAC values in L1 and the relative risk values in L2. Then use the STATPLOT feature to verify that the scatterplot follows the exponential pattern shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_001\">(Figure)<\/a>:\n<div id=\"CNX_Precalc_Figure_04_08_001\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141204\/CNX_Precalc_Figure_04_08_001.jpg\" alt=\"Graph of a scattered plot.\" width=\"487\" height=\"475\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1300931\">Use the \u201cExpReg\u201d command from the STAT then CALC menu to obtain the exponential model,<\/p>\n<div id=\"eip-id1165134361347\" class=\"unnumbered\">[latex]y=0.58304829{\\left(2.20720213\\text{E}10\\right)}^{x}[\/latex]<\/div>\n<p id=\"fs-id1370824\">Converting from scientific notation, we have:<\/p>\n<div id=\"eip-id1165134129943\" class=\"unnumbered\">[latex]y=0.58304829{\\left(\\text{22,072,021,300}\\right)}^{x}[\/latex]<\/div>\n<p id=\"fs-id1327158\">Notice that[latex]\\,{r}^{2}\\approx 0.97\\,[\/latex]which indicates the model is a good fit to the data. To see this, graph the model in the same window as the scatterplot to verify it is a good fit as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_002\">(Figure)<\/a>:<\/p>\n<div id=\"CNX_Precalc_Figure_04_08_002\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141206\/CNX_Precalc_Figure_04_08_002.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"487\" height=\"475\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/li>\n<li>\n<p id=\"fs-id1598603\">Use the model to estimate the risk associated with a BAC of[latex]\\,0.16.\\,[\/latex]Substitute[latex]\\,0.16\\,[\/latex]for[latex]\\,x\\,[\/latex]in the model and solve for[latex]\\,y.[\/latex]<\/p>\n<div id=\"eip-id1165137430511\" class=\"unnumbered\">[latex]\\begin{array}{lll}y\\hfill & =0.58304829{\\left(\\text{22,072,021,300}\\right)}^{x}\\hfill & \\text{Use the regression model found in part (a)}\\text{.}\\hfill \\\\ \\hfill & =0.58304829{\\left(\\text{22,072,021,300}\\right)}^{0.16}\\hfill & \\text{Substitute 0}\\text{.16 for }x\\text{.}\\hfill \\\\ \\hfill & \\approx \\text{26}\\text{.35}\\hfill & \\text{Round to the nearest hundredth}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<p>If a 160-pound person drives after having 6 drinks, he or she is about 26.35 times more likely to crash than if driving while sober.<\/details>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1530201\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_08_01\">\n<div id=\"fs-id1433554\">\n<p id=\"fs-id1345853\"><a class=\"autogenerated-content\" href=\"#Table_04_08_02\">(Figure)<\/a> shows a recent graduate\u2019s credit card balance each month after graduation.<\/p>\n<table id=\"Table_04_08_02\" summary=\"Two rows and ten columns. The first row is labeled, \u201cMonth\u201d, and the second row is labeled, \u201cDebt (\ud83d\udcb2)\u201d. Reading the columns as ordered pairs, we have the following values: (1, 620.00), (2, 761.88), (3, 899.80), (4, 1039.93), (5, 1270.63), (6, 1589.04), (7, 1851.31), and (8, 2154.92).\">\n<tbody>\n<tr>\n<td><strong>Month<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>Debt ($)<\/strong><\/td>\n<td>620.00<\/td>\n<td>761.88<\/td>\n<td>899.80<\/td>\n<td>1039.93<\/td>\n<td>1270.63<\/td>\n<td>1589.04<\/td>\n<td>1851.31<\/td>\n<td>2154.92<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1420469\" type=\"a\">\n<li>Use exponential regression to fit a model to these data.<\/li>\n<li>If spending continues at this rate, what will the graduate\u2019s credit card debt be one year after graduating?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1424097\" type=\"a\">\n<li>The exponential regression model that fits these data is[latex]\\,y=522.88585984{\\left(1.19645256\\right)}^{x}.[\/latex]<\/li>\n<li>If spending continues at this rate, the graduate\u2019s credit card debt will be $4,499.38 after one year.<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1395488\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1950428\"><strong>Is it reasonable to assume that an exponential regression model will represent a situation indefinitely?<\/strong><\/p>\n<p id=\"fs-id1693939\"><em>No. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation).<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1137429\" class=\"bc-section section\">\n<h3>Building a Logarithmic Model from Data<\/h3>\n<p id=\"fs-id1638611\">Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the <em>way<\/em> they increase or decrease that helps us determine whether a <span class=\"no-emphasis\">logarithmic model<\/span> is best.<\/p>\n<p id=\"fs-id1294851\">Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we\u2019ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic <span class=\"no-emphasis\">regression analysis<\/span>, we use the form of the logarithmic function most commonly used on graphing utilities,[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right).\\,[\/latex]For this function<\/p>\n<ul id=\"fs-id1505796\">\n<li>All input values,[latex]\\,x,[\/latex]must be greater than zero.<\/li>\n<li>The point[latex]\\,\\left(1,a\\right)\\,[\/latex]is on the graph of the model.<\/li>\n<li>If[latex]\\,b>0,[\/latex]the model is increasing. Growth increases rapidly at first and then steadily slows over time.<\/li>\n<li>If[latex]\\,b<0,[\/latex]the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.<\/li>\n<\/ul>\n<div id=\"fs-id1675578\" class=\"textbox key-takeaways\">\n<h3>Logarithmic Regression<\/h3>\n<p id=\"fs-id882689\"><em>Logarithmic regression<\/em> is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command \u201cLnReg\u201d on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form,<\/p>\n<div id=\"eip-id1165132974342\" class=\"unnumbered\">[latex]y=a+b\\mathrm{ln}\\left(x\\right)[\/latex]<\/div>\n<p id=\"fs-id1638058\">Note that<\/p>\n<ul id=\"fs-id934921\">\n<li>all input values,[latex]\\,x,[\/latex]must be non-negative.<\/li>\n<li>when[latex]\\,b>0,[\/latex]the model is increasing.<\/li>\n<li>when[latex]\\,b<0,[\/latex]the model is decreasing.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1530388\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1395706\"><strong>Given a set of data, perform logarithmic regression using a graphing utility.<\/strong><\/p>\n<ol id=\"fs-id1616209\" type=\"1\">\n<li>Use the STAT then EDIT menu to enter given data.\n<ol id=\"fs-id1528962\" type=\"a\">\n<li>Clear any existing data from the lists.<\/li>\n<li>List the input values in the L1 column.<\/li>\n<li>List the output values in the L2 column.<\/li>\n<\/ol>\n<\/li>\n<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\n<ol id=\"fs-id882304\" type=\"a\">\n<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\n<li>Verify the data follow a logarithmic pattern.<\/li>\n<\/ol>\n<\/li>\n<li>Find the equation that models the data.\n<ol id=\"fs-id1107843\" type=\"a\">\n<li>Select \u201cLnReg\u201d from the STAT then CALC menu.<\/li>\n<li>Use the values returned for <em>a<\/em> and <em>b<\/em> to record the model,[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right).[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_08_02\" class=\"textbox examples\">\n<div id=\"fs-id1616172\">\n<div id=\"fs-id1586202\">\n<h3>Using Logarithmic Regression to Fit a Model to Data<\/h3>\n<p id=\"fs-id1675089\">Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20th century.<\/p>\n<p id=\"eip-id1165134068998\"><a class=\"autogenerated-content\" href=\"#Table_04_08_03\">(Figure)<\/a> shows the average life expectancies, in years, of Americans from 1900\u20132010<a class=\"footnote\" title=\"\u2022Source: Center for Disease Control and Prevention, 2013\" id=\"return-footnote-112-2\" href=\"#footnote-112-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a> .<\/p>\n<table id=\"Table_04_08_03\" summary=\"Two rows and twelve columns. The first row is labeled, \u201cYear\u201d, and the second row is labeled, \u201cLife Expectancy (Years)\u201d. Reading the columns as ordered pairs, we have the following values: (1900, 47.3), (1910, 50.0), (1920, 54.1), (1930, 59.7), (1940, 62.9), (1950, 68.2), (1960, 69.7), (1970, 70.8), (1980, 73,7), (1990, 75.4), (2000, 76.8) and (2010, 78.7).\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>1900<\/td>\n<td>1910<\/td>\n<td>1920<\/td>\n<td>1930<\/td>\n<td>1940<\/td>\n<td>1950<\/td>\n<\/tr>\n<tr>\n<td><strong>Life Expectancy(Years)<\/strong><\/td>\n<td>47.3<\/td>\n<td>50.0<\/td>\n<td>54.1<\/td>\n<td>59.7<\/td>\n<td>62.9<\/td>\n<td>68.2<\/td>\n<\/tr>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>1960<\/td>\n<td>1970<\/td>\n<td>1980<\/td>\n<td>1990<\/td>\n<td>2000<\/td>\n<td>2010<\/td>\n<\/tr>\n<tr>\n<td><strong>Life Expectancy(Years)<\/strong><\/td>\n<td>69.7<\/td>\n<td>70.8<\/td>\n<td>73.7<\/td>\n<td>75.4<\/td>\n<td>76.8<\/td>\n<td>78.7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id899809\" type=\"a\">\n<li>Let[latex]\\,x\\,[\/latex]represent time in decades starting with[latex]\\,x=1\\,[\/latex]for the year 1900,[latex]\\,x=2\\,[\/latex]for the year 1910, and so on. Let[latex]\\,y\\,[\/latex]represent the corresponding life expectancy. Use logarithmic regression to fit a model to these data.<\/li>\n<li>Use the model to predict the average American life expectancy for the year 2030.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1601326\" type=\"a\">\n<li>Using the STAT then EDIT menu on a graphing utility, list the years using values 1\u201312 in L1 and the corresponding life expectancy in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logarithmic pattern as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_003\">(Figure)<\/a>:\n<div id=\"CNX_Precalc_Figure_04_08_003\" class=\"medium aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141215\/CNX_Precalc_Figure_04_08_003.jpg\" alt=\"Graph of a scattered plot.\" width=\"731\" height=\"437\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1381567\">Use the \u201cLnReg\u201d command from the STAT then CALC menu to obtain the logarithmic model,<\/p>\n<div id=\"eip-id1165137898768\" class=\"unnumbered\">[latex]y=42.52722583+13.85752327\\mathrm{ln}\\left(x\\right)[\/latex]<\/div>\n<p id=\"fs-id1677824\">Next, graph the model in the same window as the scatterplot to verify it is a good fit as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_004\">(Figure)<\/a>:<\/p>\n<div id=\"CNX_Precalc_Figure_04_08_004\" class=\"medium aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141222\/CNX_Precalc_Figure_04_08_004.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"731\" height=\"440\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/li>\n<li>To predict the life expectancy of an American in the year 2030, substitute[latex]\\,x=14\\,[\/latex]for the in the model and solve for[latex]\\,y:[\/latex]\n<div id=\"eip-id1165132035969\" class=\"unnumbered\">[latex]\\begin{array}{lll}y\\hfill & =42.52722583+13.85752327\\mathrm{ln}\\left(x\\right)\\hfill & \\text{Use the regression model found in part (a)}\\text{.}\\hfill \\\\ \\hfill & =42.52722583+13.85752327\\mathrm{ln}\\left(14\\right)\\hfill & \\text{Substitute 14 for }x\\text{.}\\hfill \\\\ \\hfill & \\approx \\text{79}\\text{.1}\\hfill & \\text{Round to the nearest tenth.}\\hfill \\end{array}[\/latex]<\/div>\n<p>If life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the year 2030.<\/details>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1338539\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_08_02\">\n<div id=\"fs-id1527553\">\n<p id=\"fs-id899893\">Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. <a class=\"autogenerated-content\" href=\"#Table_04_08_04\">(Figure)<\/a> shows the number of games sold, in thousands, from the years 2000\u20132010.<\/p>\n<table id=\"Table_04_08_04\" summary=\"Two rows and twelve columns. The first row is labeled, \u201cYear\u201d, and the second row is labeled, \u201cNumber Sold (Thousands)\u201d. Reading the columns as ordered pairs, we have the following values: (2000, 142), (2001, 149), (2002, 154), (2003, 155), (2004, 159), (2005, 161), (2006, 163), (2007, 164), (2008, 164), (2009, 166), and (2010, 167).\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2000<\/td>\n<td>2001<\/td>\n<td>2002<\/td>\n<td>2003<\/td>\n<td>2004<\/td>\n<td>2005<\/td>\n<\/tr>\n<tr>\n<td><strong>Number Sold (thousands)<\/strong><\/td>\n<td>142<\/td>\n<td>149<\/td>\n<td>154<\/td>\n<td>155<\/td>\n<td>159<\/td>\n<td>161<\/td>\n<\/tr>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<tr>\n<td><strong>Number Sold (thousands)<\/strong><\/td>\n<td>163<\/td>\n<td>164<\/td>\n<td>164<\/td>\n<td>166<\/td>\n<td>167<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1420568\" type=\"a\">\n<li>Let[latex]\\,x\\,[\/latex]represent time in years starting with[latex]\\,x=1\\,[\/latex]for the year 2000. Let[latex]\\,y\\,[\/latex]represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data.<\/li>\n<li>If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1365229\" type=\"a\">\n<li>The logarithmic regression model that fits these data is[latex]\\,y=141.91242949+10.45366573\\mathrm{ln}\\left(x\\right)\\,[\/latex]<\/li>\n<li>If sales continue at this rate, about 171,000 games will be sold in the year 2015.<\/details>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1599403\" class=\"bc-section section\">\n<h3>Building a Logistic Model from Data<\/h3>\n<p id=\"fs-id1316523\">Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or <em>limiting value<\/em>. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.<\/p>\n<p id=\"fs-id1677718\">It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic <span class=\"no-emphasis\">regression analysis<\/span>, we use the form most commonly used on graphing utilities:<\/p>\n<div id=\"eip-154\" class=\"unnumbered aligncenter\">[latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex]<\/div>\n<p id=\"fs-id1310104\">Recall that:<\/p>\n<ul id=\"fs-id1294720\">\n<li>[latex]\\frac{c}{1+a}\\,[\/latex]is the initial value of the model.<\/li>\n<li>when[latex]\\,b>0,[\/latex] the model increases rapidly at first until it reaches its point of maximum growth rate,[latex]\\,\\left(\\frac{\\mathrm{ln}\\left(a\\right)}{b},\\frac{c}{2}\\right).\\,[\/latex]At that point, growth steadily slows and the function becomes asymptotic to the upper bound[latex]\\,y=c.[\/latex]<\/li>\n<li>[latex]c\\,[\/latex]<br \/>\nis the limiting value, sometimes called the <em>carrying capacity<\/em>, of the model.<\/li>\n<\/ul>\n<div id=\"fs-id1454974\" class=\"textbox key-takeaways\">\n<h3>Logistic Regression<\/h3>\n<p id=\"fs-id1583226\"><em>Logistic regression<\/em> is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command \u201cLogistic\u201d on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form<\/p>\n<div class=\"unnumbered\">[latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex]<\/div>\n<p id=\"fs-id1701600\">Note that<\/p>\n<ul id=\"fs-id1358262\">\n<li>The initial value of the model is[latex]\\,\\frac{c}{1+a}.[\/latex]<\/li>\n<li>Output values for the model grow closer and closer to[latex]\\,y=c\\,[\/latex]as time increases.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1361145\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1676582\"><strong>Given a set of data, perform logistic regression using a graphing utility.<\/strong><\/p>\n<ol id=\"fs-id1690756\" type=\"1\">\n<li>Use the STAT then EDIT menu to enter given data.\n<ol id=\"fs-id1562359\" type=\"a\">\n<li>Clear any existing data from the lists.<\/li>\n<li>List the input values in the L1 column.<\/li>\n<li>List the output values in the L2 column.<\/li>\n<\/ol>\n<\/li>\n<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\n<ol id=\"fs-id1431005\" type=\"a\">\n<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\n<li>Verify the data follow a logistic pattern.<\/li>\n<\/ol>\n<\/li>\n<li>Find the equation that models the data.\n<ol id=\"fs-id1585412\" type=\"a\">\n<li>Select \u201cLogistic\u201d from the STAT then CALC menu.<\/li>\n<li>Use the values returned for[latex]\\,a,[\/latex][latex]\\,b,[\/latex] and[latex]\\,c\\,[\/latex]to record the model,[latex]\\,y=\\frac{c}{1+a{e}^{-bx}}.[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_08_03\" class=\"textbox examples\">\n<div id=\"fs-id1646727\">\n<div id=\"fs-id1523395\">\n<h3>Using Logistic Regression to Fit a Model to Data<\/h3>\n<p id=\"fs-id1422087\">Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. <a class=\"autogenerated-content\" href=\"#Table_04_08_05\">(Figure)<\/a> shows the percentage of Americans with cellular service between the years 1995 and 2012<a class=\"footnote\" title=\"\u2022Source: The World Bank, 2013\" id=\"return-footnote-112-3\" href=\"#footnote-112-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a> .<\/p>\n<table id=\"Table_04_08_05\" summary=\"Nineteen rows and two columns. The first column is labeled, \u201cYear\u201d, and the second column is labeled, \u201cAmericans with Cellular Service (%)\u201d. Reading the columns as ordered pairs, we have the following values: (1995, 12.69), (1996, 16.35), (1997, 20.29), (1998, 25.08), (1999, 30.81), (2000, 38.75), (2001, 45.00), (2002, 49.16), (2003, 55.15), (2004, 62.85), (2005, 68.63), (2006, 76.64), (2007, 82.47), (2008, 85.68), (2009, 89.14), (2010, 91.86), (2011, 95.28), and (2012, 98.17).\">\n<thead>\n<tr>\n<th>Year<\/th>\n<th>Americans with Cellular Service (%)<\/th>\n<th>Year<\/th>\n<th>Americans with Cellular Service (%)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1995<\/td>\n<td>12.69<\/td>\n<td>2004<\/td>\n<td>62.852<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>16.35<\/td>\n<td>2005<\/td>\n<td>68.63<\/td>\n<\/tr>\n<tr>\n<td>1997<\/td>\n<td>20.29<\/td>\n<td>2006<\/td>\n<td>76.64<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>25.08<\/td>\n<td>2007<\/td>\n<td>82.47<\/td>\n<\/tr>\n<tr>\n<td>1999<\/td>\n<td>30.81<\/td>\n<td>2008<\/td>\n<td>85.68<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>38.75<\/td>\n<td>2009<\/td>\n<td>89.14<\/td>\n<\/tr>\n<tr>\n<td>2001<\/td>\n<td>45.00<\/td>\n<td>2010<\/td>\n<td>91.86<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>49.16<\/td>\n<td>2011<\/td>\n<td>95.28<\/td>\n<\/tr>\n<tr>\n<td>2003<\/td>\n<td>55.15<\/td>\n<td>2012<\/td>\n<td>98.17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1624806\" type=\"a\">\n<li>Let[latex]\\,x\\,[\/latex]represent time in years starting with[latex]\\,x=0\\,[\/latex]for the year 1995. Let[latex]\\,y\\,[\/latex]represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data.<\/li>\n<li>Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent.<\/li>\n<li>Discuss the value returned for the upper limit,[latex]\\,c.\\,[\/latex]What does this tell you about the model? What would the limiting value be if the model were exact?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1455761\" type=\"a\">\n<li>Using the STAT then EDIT menu on a graphing utility, list the years using values 0\u201315 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_005\">(Figure)<\/a>:\n<div id=\"CNX_Precalc_Figure_04_08_005\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141225\/CNX_Precalc_Figure_04_08_005.jpg\" alt=\"Graph of a scattered plot.\" width=\"975\" height=\"479\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1366003\">Use the \u201cLogistic\u201d command from the STAT then CALC menu to obtain the logistic model,<\/p>\n<div id=\"eip-id1165134104896\" class=\"unnumbered\">[latex]y=\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}[\/latex]<\/div>\n<p id=\"fs-id1650001\">Next, graph the model in the same window as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_08_006\">(Figure)<\/a> the scatterplot to verify it is a good fit:<\/p>\n<div id=\"CNX_Precalc_Figure_04_08_006\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141228\/CNX_Precalc_Figure_04_08_006.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"975\" height=\"479\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/li>\n<li>\n<p id=\"fs-id1410550\">To approximate the percentage of Americans with cellular service in the year 2013, substitute[latex]\\,x=18\\,[\/latex]for the in the model and solve for[latex]\\,y:[\/latex]<\/p>\n<div id=\"eip-9\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{lll}y\\hfill & =\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}\\hfill & \\text{Use the regression model found in part (a)}.\\hfill \\\\ \\hfill & =\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013\\left(18\\right)}}\\hfill & \\text{Substitute 18 for }x.\\hfill \\\\ \\hfill & \\approx \\text{99}\\text{.3 }\\hfill & \\text{Round to the nearest tenth}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1569617\">According to the model, about 98.8% of Americans had cellular service in 2013.<\/p>\n<\/li>\n<li>The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be[latex]\\,c=100\\,[\/latex]and the model\u2019s outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service!<\/details>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1394388\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_08_03\">\n<div id=\"fs-id1521431\">\n<p id=\"fs-id1638072\"><a class=\"autogenerated-content\" href=\"#Table_04_08_06\">(Figure)<\/a> shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012.<\/p>\n<table id=\"Table_04_08_06\" summary=\"Seventeen rows and two columns. The first column is labeled, \u201cYear\u201d, and the second column is labeled, \u201cSeal Population (Thousands)\u201d. Reading the columns as ordered pairs, we have the following values: (1997, 3.493), (1998, 5.282), (1999, 6.357), (2000, 9.201), (2001, 11.224), (2002, 12.964), (2003, 16.226), (2004, 18.137), (2005, 19.590), (2006, 21.955), (2007, 22.862), (2008, 23.869), (2009, 24.243), (2010, 24.344), (2011, 24.919), and (2012, 25.108).\">\n<thead>\n<tr>\n<th>Year<\/th>\n<th>Seal Population (Thousands)<\/th>\n<th>Year<\/th>\n<th>Seal Population (Thousands)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1997<\/td>\n<td>3.493<\/td>\n<td>2005<\/td>\n<td>19.590<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>5.282<\/td>\n<td>2006<\/td>\n<td>21.955<\/td>\n<\/tr>\n<tr>\n<td>1999<\/td>\n<td>6.357<\/td>\n<td>2007<\/td>\n<td>22.862<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>9.201<\/td>\n<td>2008<\/td>\n<td>23.869<\/td>\n<\/tr>\n<tr>\n<td>2001<\/td>\n<td>11.224<\/td>\n<td>2009<\/td>\n<td>24.243<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>12.964<\/td>\n<td>2010<\/td>\n<td>24.344<\/td>\n<\/tr>\n<tr>\n<td>2003<\/td>\n<td>16.226<\/td>\n<td>2011<\/td>\n<td>24.919<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>18.137<\/td>\n<td>2012<\/td>\n<td>25.108<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1366862\" type=\"a\">\n<li>Let[latex]\\,x\\,[\/latex]represent time in years starting with[latex]\\,x=0\\,[\/latex]for the year 1997. Let[latex]\\,y\\,[\/latex]represent the number of seals in thousands. Use logistic regression to fit a model to these data.<\/li>\n<li>Use the model to predict the seal population for the year 2020.<\/li>\n<li>To the nearest whole number, what is the limiting value of this model?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1609961\" type=\"a\">\n<li>The logistic regression model that fits these data is[latex]\\,y=\\frac{25.65665979}{1+6.113686306{e}^{-0.3852149008x}}.[\/latex]<\/li>\n<li>If the population continues to grow at this rate, there will be about[latex]\\,\\text{25,634}\\,[\/latex]seals in 2020.<\/li>\n<li>To the nearest whole number, the carrying capacity is 25,657.<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1384374\" class=\"precalculus media\">\n<p id=\"fs-id1676985\">Access this online resource for additional instruction and practice with exponential function models.<\/p>\n<ul id=\"fs-id1676988\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/pregresscalc\">Exponential Regression on a Calculator<\/a><\/li>\n<\/ul>\n<\/div>\n<p id=\"eip-136\">Visit <a href=\"http:\/\/openstaxcollege.org\/l\/PreCalcLPC04\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n<\/div>\n<div id=\"fs-id1294229\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1410554\">\n<li>Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.<\/li>\n<li>We use the command \u201cExpReg\u201d on a graphing utility to fit function of the form[latex]\\,y=a{b}^{x}\\,[\/latex]to a set of data points. See <a class=\"autogenerated-content\" href=\"#Example_04_08_01\">(Figure)<\/a>.<\/li>\n<li>Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.<\/li>\n<li>We use the command \u201cLnReg\u201d on a graphing utility to fit a function of the form[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right)\\,[\/latex]to a set of data points. See <a class=\"autogenerated-content\" href=\"#Example_04_08_02\">(Figure)<\/a>.<\/li>\n<li>Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.<\/li>\n<li>We use the command \u201cLogistic\u201d on a graphing utility to fit a function of the form[latex]\\,y=\\frac{c}{1+a{e}^{-bx}}\\,[\/latex]to a set of data points. See <a class=\"autogenerated-content\" href=\"#Example_04_08_03\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1700714\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1700718\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1402840\">\n<div id=\"fs-id1402843\">\n<p id=\"fs-id1424627\">What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1660264\">Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1692586\">\n<div id=\"fs-id1692588\">\n<p id=\"fs-id1033498\">What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1447042\">\n<div id=\"fs-id1523478\">\n<p id=\"fs-id1523480\">What is regression analysis? Describe the process of performing regression analysis on a graphing utility.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1648021\">Regression analysis is the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the given points using the STAT then EDIT menu. Next graph the scatter plot using the STAT PLOT feature. The shape of the data points on the scatter graph can help determine which regression feature to use. Once this is determined, select the appropriate regression analysis command from the STAT then CALC menu.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1562322\">\n<div id=\"fs-id1562324\">\n<p id=\"fs-id1271600\">What might a scatterplot of data points look like if it were best described by a logarithmic model?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1638051\">\n<div id=\"fs-id1638053\">\n<p id=\"fs-id916950\">What does the <em>y<\/em>-intercept on the graph of a logistic equation correspond to for a population modeled by that equation?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1586837\">The <em>y<\/em>-intercept on the graph of a logistic equation corresponds to the initial population for the population model.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1457107\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1404306\">For the following exercises, match the given function of best fit with the appropriate scatterplot in <a class=\"autogenerated-content\" href=\"#CNX_PreCalc_Figure_04_08_201\">(Figure)<\/a> through <a class=\"autogenerated-content\" href=\"#CNX_PreCalc_Figure_04_08_205\">(Figure)<\/a><strong>. <\/strong>Answer using the letter beneath the matching graph.<\/p>\n<div id=\"CNX_Precalc_Figure_04_08_201\" class=\"small aligncenter\"><span id=\"fs-id299196\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141230\/CNX_PreCalc_Figure_04_08_201.jpg\" alt=\"Graph of a scattered plot.\" \/><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_04_08_202\" class=\"small aligncenter\"><span id=\"fs-id1425423\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141232\/CNX_PreCalc_Figure_04_08_202.jpg\" alt=\"Graph of a scattered plot.\" \/><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_04_08_203\" class=\"small aligncenter\"><span id=\"fs-id1200734\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141240\/CNX_PreCalc_Figure_04_08_203.jpg\" alt=\"Graph of a scattered plot.\" \/><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_04_08_204\" class=\"small aligncenter\"><span id=\"fs-id1692902\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141243\/CNX_PreCalc_Figure_04_08_204.jpg\" alt=\"Graph of a scattered plot.\" \/><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_04_08_205\" class=\"small aligncenter\"><span id=\"fs-id1425975\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141253\/CNX_PreCalc_Figure_04_08_205.jpg\" alt=\"Graph of a scattered plot.\" \/><\/span><\/div>\n<div id=\"fs-id1670879\">\n<div id=\"fs-id1670881\">\n<p id=\"fs-id1700443\">[latex]y=10.209{e}^{-0.294x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1701699\">\n<div id=\"fs-id1701701\">\n<p id=\"fs-id1428638\">[latex]y=5.598-1.912\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1423720\">C<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1519709\">\n<div id=\"fs-id1519711\">\n<p id=\"fs-id1422545\">[latex]y=2.104{\\left(1.479\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1693326\">\n<div id=\"fs-id1692337\">\n<p id=\"fs-id1692340\">[latex]y=4.607+2.733\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1696917\">B<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1410055\">\n<div id=\"fs-id1385282\">\n<p id=\"fs-id1385284\">[latex]y=\\frac{14.005}{1+2.79{e}^{-0.812x}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1659453\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<div id=\"fs-id1522155\">\n<div id=\"fs-id1522157\">\n<p id=\"fs-id1522159\">To the nearest whole number, what is the initial value of a population modeled by the logistic equation[latex]\\,P\\left(t\\right)=\\frac{175}{1+6.995{e}^{-0.68t}}?\\,[\/latex]What is the carrying capacity?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1661536\">[latex]P\\left(0\\right)=22\\,[\/latex]; 175<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1346082\">\n<div id=\"fs-id1346085\">\n<p id=\"fs-id1660230\">Rewrite the exponential model[latex]\\,A\\left(t\\right)=1550{\\left(1.085\\right)}^{x}\\,[\/latex]as an equivalent model with base[latex]\\,e.\\,[\/latex]Express the exponent to four significant digits.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1697446\">\n<div id=\"fs-id1675823\">\n<p id=\"fs-id1675825\">A logarithmic model is given by the equation[latex]\\,h\\left(p\\right)=67.682-5.792\\mathrm{ln}\\left(p\\right).\\,[\/latex]To the nearest hundredth, for what value of[latex]\\,p\\,[\/latex]does[latex]\\,h\\left(p\\right)=62?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1086031\">[latex]p\\approx 2.67[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1405194\">\n<div id=\"fs-id1405196\">\n<p id=\"fs-id1381601\">A logistic model is given by the equation[latex]\\,P\\left(t\\right)=\\frac{90}{1+5{e}^{-0.42t}}.\\,[\/latex]To the nearest hundredth, for what value of <em>t<\/em> does[latex]\\,P\\left(t\\right)=45?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1439754\">\n<div id=\"fs-id1439756\">\n<p id=\"fs-id1569418\">What is the <em>y<\/em>-intercept on the graph of the logistic model given in the previous exercise?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1697176\"><em>y<\/em>-intercept:[latex]\\,\\left(0,15\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1702994\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1422233\">For the following exercises, use this scenario: The population[latex]\\,P\\,[\/latex]of a koi pond over[latex]\\,x\\,[\/latex]months is modeled by the function[latex]\\,P\\left(x\\right)=\\frac{68}{1+16{e}^{-0.28x}}.[\/latex]<\/p>\n<div id=\"fs-id1246586\">\n<div id=\"fs-id1246589\">\n<p id=\"fs-id1457169\">Graph the population model to show the population over a span of[latex]\\,3\\,[\/latex]years.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1408601\">\n<div id=\"fs-id1408603\">\n<p id=\"fs-id1408605\">What was the initial population of koi?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1440059\">[latex]4\\,[\/latex]koi<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1535423\">\n<div id=\"fs-id1535425\">\n<p id=\"fs-id1678373\">How many koi will the pond have after one and a half years?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1678378\">\n<div id=\"fs-id1423485\">\n<p id=\"fs-id1423487\">How many months will it take before there are[latex]\\,20\\,[\/latex]koi in the pond?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1394674\">about[latex]\\,6.8\\,[\/latex]months.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1582861\">\n<div id=\"fs-id1420623\">\n<p id=\"fs-id1420625\">Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity.<\/p>\n<\/div>\n<div id=\"eip-id2711495\">\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141255\/CNX_Precalc_Figure_04_08_207.jpg\" alt=\"Graph of the intersection of P(t)=68\/(1+16e^(-0.28t)) and y=34.\" \/><\/details>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1706412\">For the following exercises, use this scenario: The population[latex]\\,P\\,[\/latex]of an endangered species habitat for wolves is modeled by the function[latex]\\,P\\left(x\\right)=\\frac{558}{1+54.8{e}^{-0.462x}},[\/latex] where[latex]\\,x\\,[\/latex]is given in years.<\/p>\n<div id=\"fs-id1658971\">\n<div id=\"fs-id1658973\">\n<p id=\"fs-id1658975\">Graph the population model to show the population over a span of[latex]\\,10\\,[\/latex]years.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1701908\">\n<div id=\"fs-id1504971\">\n<p id=\"fs-id1504973\">What was the initial population of wolves transported to the habitat?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1409591\">[latex]10\\,[\/latex]wolves<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1505494\">\n<div id=\"fs-id1505496\">\n<p id=\"fs-id1505498\">How many wolves will the habitat have after[latex]\\,3\\,[\/latex]years?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1271453\">\n<div id=\"fs-id925282\">\n<p id=\"fs-id925284\">How many years will it take before there are[latex]\\,100\\,[\/latex]wolves in the habitat?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1459669\">about 5.4 years.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1601914\">\n<div id=\"fs-id1601916\">\n<p id=\"fs-id1523061\">Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1597787\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_07\">(Figure)<\/a>.<\/p>\n<table id=\"Table_04_08_07\" summary=\"Two columns and seven row. The first column labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 1125), (2, 1495), (3, 2310), (4, 3295), (5, 4650), and (6, 6361).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>1125<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>1495<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>2310<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>3294<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>4650<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>6361<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1432918\">\n<div id=\"fs-id1699848\">\n<p id=\"fs-id1699850\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1803598\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141259\/CNX_PreCalc_Figure_04_08_210.jpg\" alt=\"Graph of the table\u2019s values.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1514386\">\n<div id=\"fs-id1648035\">\n<p id=\"fs-id1648037\">Use the regression feature to find an exponential function that best fits the data in the table.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1586463\">\n<div id=\"fs-id1586466\">\n<p id=\"fs-id1597729\">Write the exponential function as an exponential equation with base[latex]\\,e.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1600227\">[latex]f\\left(x\\right)=776.682{e}^{0.3549x}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1447084\">\n<div id=\"fs-id1447087\">\n<p id=\"fs-id1447089\">Graph the exponential equation on the scatter diagram.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1447005\">\n<div id=\"fs-id1376767\">\n<p id=\"fs-id1376769\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which[latex]\\,f\\left(x\\right)=4000.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1221397\">When[latex]\\,f\\left(x\\right)=4000,[\/latex][latex]x\\approx 4.6.[\/latex]<\/p>\n<p><span id=\"fs-id1436308\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141306\/CNX_PreCalc_Figure_04_08_212.jpg\" alt=\"Graph of the intersection of a scattered plot with an estimation line and y=4,000.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1511970\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_08\">(Figure)<\/a>.<\/p>\n<table id=\"Table_04_08_08\" summary=\"Two columns and seven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 555), (2, 383), (3, 307), (4, 210), (5, 158), and (6, 122).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>555<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>383<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>307<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>210<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>158<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>122<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1407921\">\n<div id=\"fs-id1407923\">\n<p id=\"fs-id1407925\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1671727\">\n<div id=\"fs-id1692964\">\n<p id=\"fs-id1692966\">Use the regression feature to find an exponential function that best fits the data in the table.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1405103\">[latex]f\\left(x\\right)=731.92{\\left(0.738\\right)}^{x}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1652039\">\n<div id=\"fs-id1652041\">\n<p id=\"fs-id1652043\">Write the exponential function as an exponential equation with base[latex]\\,e.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1590772\">\n<div id=\"fs-id1513210\">\n<p id=\"fs-id1513213\">Graph the exponential equation on the scatter diagram.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1435907\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141309\/CNX_PreCalc_Figure_04_08_214.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1606195\">\n<div id=\"fs-id1424979\">\n<p id=\"fs-id1424981\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which[latex]\\,f\\left(x\\right)=250.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1460953\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_09\">(Figure)<\/a>.<\/p>\n<table id=\"Table_04_08_09\" summary=\"Two columns and seven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 5.1), (2, 6.3), (3, 7.3), (4, 7.7), (5, 8.1), and (6, 8.6).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>5.1<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>6.3<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>7.3<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>7.7<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>8.1<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>8.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1371063\">\n<div id=\"fs-id1371065\">\n<p id=\"fs-id1371067\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1588274\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141312\/CNX_PreCalc_Figure_04_08_216.jpg\" alt=\"Graph of the table\u2019s values.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1650282\">\n<div id=\"fs-id1650285\">\n<p id=\"fs-id1420307\">Use the LOGarithm option of the REGression feature to find a logarithmic function of the form[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right)\\,[\/latex]that best fits the data in the table.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1432678\">\n<div id=\"fs-id1432680\">\n<p id=\"fs-id1356158\">Use the logarithmic function to find the value of the function when[latex]\\,x=10.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1518348\">[latex]f\\left(10\\right)\\approx 9.5[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1392056\">\n<div id=\"fs-id1392058\">\n<p id=\"fs-id1392060\">Graph the logarithmic equation on the scatter diagram.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1649251\">\n<div id=\"fs-id1649253\">\n<p id=\"fs-id1355295\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which[latex]\\,f\\left(x\\right)=7.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1434551\">When[latex]\\,f\\left(x\\right)=7,[\/latex] [latex]x\\approx 2.7.[\/latex]<\/p>\n<p><span id=\"fs-id1381504\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141314\/CNX_PreCalc_Figure_04_08_218.jpg\" alt=\"Graph of the intersection of a scattered plot with an estimation line and y=7.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1694168\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_10\">(Figure)<\/a>.<\/p>\n<table id=\"Table_04_08_10\" summary=\"Two columns and nine nows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 7.5), (2, 6), (3, 5.2), (4, 4.3), (5, 3.9), (6, 3.4), (7, 3.1), and (8, 2.9).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>7.5<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>5.2<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>4.3<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>3.9<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>3.4<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>3.1<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>2.9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1432065\">\n<div id=\"fs-id1432067\">\n<p id=\"fs-id1432069\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1426060\">\n<div id=\"fs-id1426062\">\n<p id=\"fs-id1426064\">Use the LOGarithm option of the REGression feature to find a logarithmic function of the form[latex]\\,y=a+b\\mathrm{ln}\\left(x\\right)\\,[\/latex]that best fits the data in the table.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1393286\">[latex]f\\left(x\\right)=7.544-2.268\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1424700\">\n<div id=\"fs-id1601388\">\n<p id=\"fs-id1601390\">Use the logarithmic function to find the value of the function when[latex]\\,x=10.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1649327\">\n<div id=\"fs-id1649329\">\n<p id=\"fs-id1649331\">Graph the logarithmic equation on the scatter diagram.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1364738\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141320\/CNX_PreCalc_Figure_04_08_220.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1451593\">\n<div id=\"fs-id1451595\">\n<p id=\"fs-id1211247\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which[latex]\\,f\\left(x\\right)=8.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1661470\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_11\">(Figure)<\/a>.<\/p>\n<table id=\"Table_04_08_11\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 8.7), (2, 12.3), (3, 15.4), (4, 18.5), (5, 20.7), (6, 22.5), (7, 23.3), (8, 24), (9, 24.6), and (10, 24.8).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>8.7<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>12.3<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>15.4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>18.5<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>20.7<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>22.5<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>23.3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>24<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>24.6<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>24.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1656649\">\n<div id=\"fs-id1407656\">\n<p id=\"fs-id1407658\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1439003\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141323\/CNX_PreCalc_Figure_04_08_222.jpg\" alt=\"Graph of the table\u2019s values.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1423807\">\n<div>\n<p>Use the LOGISTIC regression option to find a logistic growth model of the form[latex]\\,y=\\frac{c}{1+a{e}^{-bx}}\\,[\/latex]that best fits the data in the table.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1405629\">\n<div id=\"fs-id1405631\">\n<p id=\"fs-id1671038\">Graph the logistic equation on the scatter diagram.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1523360\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141326\/CNX_Precalc_Figure_04_08_223.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1535974\">\n<div id=\"fs-id1508700\">\n<p id=\"fs-id1508702\">To the nearest whole number, what is the predicted carrying capacity of the model?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1508707\">\n<div id=\"fs-id1609403\">\n<p id=\"fs-id1609405\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which the model reaches half its carrying capacity.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1370548\">When[latex]\\,f\\left(x\\right)=12.5,[\/latex] [latex]x\\approx 2.1.[\/latex]<\/p>\n<p><span id=\"fs-id1689750\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141339\/CNX_Precalc_Figure_04_08_224.jpg\" alt=\"Graph of the intersection of a scattered plot with an estimation line and y=12.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1433707\">For the following exercises, refer to <a class=\"autogenerated-content\" href=\"#Table_04_08_12\">(Figure)<\/a>.<\/p>\n<table id=\"Table_04_08_12\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (0, 12), (2, 28.6), (4, 52.8), (5, 70.3), (7, 99.9), (8, 112.5), (10, 125.8), (11, 127.9), (15, 135.1), and (17, 135.9).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>28.6<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>52.8<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>70.3<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>99.9<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>112.5<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>125.8<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>127.9<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>135.1<\/td>\n<\/tr>\n<tr>\n<td>17<\/td>\n<td>135.9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1440074\">\n<div id=\"fs-id1440076\">\n<p id=\"fs-id1440079\">Use a graphing calculator to create a scatter diagram of the data.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1520217\">\n<div id=\"fs-id1520219\">\n<p id=\"fs-id1520221\">Use the LOGISTIC regression option to find a logistic growth model of the form[latex]\\,y=\\frac{c}{1+a{e}^{-bx}}\\,[\/latex]that best fits the data in the table.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1375601\">[latex]f\\left(x\\right)=\\frac{136.068}{1+10.324{e}^{-0.480x}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1705222\">\n<div id=\"fs-id1705224\">\n<p id=\"fs-id1705226\">Graph the logistic equation on the scatter diagram.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1531111\">\n<div id=\"fs-id1531113\">\n<p id=\"fs-id1531115\">To the nearest whole number, what is the predicted carrying capacity of the model?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1405350\">about[latex]136[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1406372\">\n<div id=\"fs-id1406374\">\n<p id=\"fs-id1406377\">Use the intersect feature to find the value of[latex]\\,x\\,[\/latex]for which the model reaches half its carrying capacity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1583097\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1583102\">\n<div id=\"fs-id1583104\">\n<p id=\"fs-id1404836\">Recall that the general form of a logistic equation for a population is given by[latex]\\,P\\left(t\\right)=\\frac{c}{1+a{e}^{-bt}},[\/latex] such that the initial population at time[latex]\\,t=0\\,[\/latex]is[latex]\\,P\\left(0\\right)={P}_{0}.\\,[\/latex]Show algebraically that[latex]\\,\\frac{c-P\\left(t\\right)}{P\\left(t\\right)}=\\frac{c-{P}_{0}}{{P}_{0}}{e}^{-bt}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1366138\">Working with the left side of the equation, we see that it can be rewritten as[latex]\\,a{e}^{-bt}:[\/latex]<\/p>\n<p id=\"fs-id1457268\">[latex]\\frac{c-P\\left(t\\right)}{P\\left(t\\right)}=\\frac{c-\\frac{c}{1+a{e}^{-bt}}}{\\frac{c}{1+a{e}^{-bt}}}=\\frac{\\frac{c\\left(1+a{e}^{-bt}\\right)-c}{1+a{e}^{-bt}}}{\\frac{c}{1+a{e}^{-bt}}}=\\frac{\\frac{c\\left(1+a{e}^{-bt}-1\\right)}{1+a{e}^{-bt}}}{\\frac{c}{1+a{e}^{-bt}}}=1+a{e}^{-bt}-1=a{e}^{-bt}[\/latex]<\/p>\n<p id=\"fs-id1460591\">Working with the right side of the equation we show that it can also be rewritten as[latex]\\,a{e}^{-bt}.\\,[\/latex]But first note that when[latex]\\,t=0,[\/latex]<br \/>\n[latex]\\,{P}_{0}=\\frac{c}{1+a{e}^{-b\\left(0\\right)}}=\\frac{c}{1+a}.\\,[\/latex]Therefore,<\/p>\n<p id=\"fs-id1283132\">[latex]\\frac{c-{P}_{0}}{{P}_{0}}{e}^{-bt}=\\frac{c-\\frac{c}{1+a}}{\\frac{c}{1+a}}{e}^{-bt}=\\frac{\\frac{c\\left(1+a\\right)-c}{1+a}}{\\frac{c}{1+a}}{e}^{-bt}=\\frac{\\frac{c\\left(1+a-1\\right)}{1+a}}{\\frac{c}{1+a}}{e}^{-bt}=\\left(1+a-1\\right){e}^{-bt}=a{e}^{-bt}[\/latex]<\/p>\n<p>Thus,[latex]\\,\\frac{c-P\\left(t\\right)}{P\\left(t\\right)}=\\frac{c-{P}_{0}}{{P}_{0}}{e}^{-bt}.[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1435401\">\n<div id=\"fs-id1435404\">\n<p id=\"fs-id1658819\">Use a graphing utility to find an exponential regression formula[latex]\\,f\\left(x\\right)\\,[\/latex]and a logarithmic regression formula[latex]\\,g\\left(x\\right)\\,[\/latex]for the points[latex]\\,\\left(1.5,1.5\\right)\\,[\/latex]and[latex]\\,\\left(8.5,\\text{ 8}\\text{.5}\\right).\\,[\/latex]Round all numbers to 6 decimal places. Graph the points and both formulas along with the line[latex]\\,y=x\\,[\/latex]on the same axis. Make a conjecture about the relationship of the regression formulas.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1583256\">\n<div id=\"fs-id1583258\">\n<p id=\"fs-id1658556\">Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1658563\">First rewrite the exponential with base <em>e<\/em>:[latex]\\,f\\left(x\\right)=1.034341{e}^{\\text{0}\\text{.247800x}}.\\,[\/latex]Then test to verify that[latex]\\,f\\left(g\\left(x\\right)\\right)=x,[\/latex]taking rounding error into consideration:<\/p>\n<p id=\"fs-id1513342\">[latex]\\begin{array}{ll}g\\left(f\\left(x\\right)\\right)\\hfill & =4.035510\\mathrm{ln}\\left(1.034341{e}^{\\text{0}\\text{.247800x}}\\,\\right)-0.136259\\hfill \\\\ \\hfill & =4.03551\\left(\\mathrm{ln}\\left(1.034341\\right)+\\mathrm{ln}\\left({e}^{\\text{0}\\text{.2478}x}\\,\\right)\\right)-0.136259\\hfill \\\\ \\hfill & =4.03551\\left(\\mathrm{ln}\\left(1.034341\\right)+\\text{0}\\text{.2478}x\\right)-0.136259\\hfill \\\\ \\hfill & =0.136257+0.999999x-0.136259\\hfill \\\\ \\hfill & =-0.000002+0.999999x\\hfill \\\\ \\hfill & \\approx 0+x\\hfill \\\\ \\hfill & =x\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1700899\">\n<div id=\"fs-id1532222\">\n<p id=\"fs-id1532224\">Find the inverse function[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]for the logistic function[latex]\\,f\\left(x\\right)=\\frac{c}{1+a{e}^{-bx}}.\\,[\/latex]Show all steps.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1631452\">\n<div id=\"fs-id1631454\">\n<p id=\"fs-id1631456\">Use the result from the previous exercise to graph the logistic model[latex]\\,P\\left(t\\right)=\\frac{20}{1+4{e}^{-0.5t}}\\,[\/latex]along with its inverse on the same axis. What are the intercepts and asymptotes of each function?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1440690\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141350\/CNX_PreCalc_Figure_04_08_228.jpg\" alt=\"Graph of P(t)=20\/(1+40.5e^(-0.5t)) and P(t)=(ln(4)-ln((20\/t)-1)\/0.5.\" \/><\/span><\/p>\n<p id=\"eip-id1165134486767\">The graph of[latex]\\,P\\left(t\\right)\\,[\/latex]has a <em>y<\/em>-intercept at (0, 4) and horizontal asymptotes at <em>y<\/em> = 0 and <em>y<\/em> = 20. The graph of[latex]\\,{P}^{-1}\\left(t\\right)\\,[\/latex]has an <em>x<\/em>&#8211; intercept at (4, 0) and vertical asymptotes at <em>x<\/em> = 0 and <em>x<\/em> = 20.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1582712\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1582715\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/694d2eea-4135-47f3-8c06-60472f7e967c\">Exponential Functions<\/a><\/h4>\n<div id=\"fs-id1705125\">\n<div id=\"fs-id1705127\">\n<p id=\"fs-id1705129\">Determine whether the function[latex]\\,y=156{\\left(0.825\\right)}^{t}\\,[\/latex]represents exponential growth, exponential decay, or neither. Explain<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1504948\">exponential decay; The growth factor,[latex]\\,0.825,[\/latex] is between[latex]\\,0\\,[\/latex]and[latex]\\,1.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1461193\">\n<div id=\"fs-id1461195\">\n<p id=\"fs-id1461197\">The population of a herd of deer is represented by the function[latex]\\,A\\left(t\\right)=205{\\left(1.13\\right)}^{t},\\,[\/latex]where[latex]\\,t\\,[\/latex]is given in years. To the nearest whole number, what will the herd population be after[latex]\\,6\\,[\/latex]years?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1658863\">\n<div id=\"fs-id1658865\">\n<p id=\"fs-id1658867\">Find an exponential equation that passes through the points[latex]\\,\\text{(2, 2}\\text{.25)}\\,[\/latex]and[latex]\\,\\left(5,60.75\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1655487\">[latex]y=0.25{\\left(3\\right)}^{x}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1394625\">\n<div id=\"fs-id1394627\">\n<p id=\"fs-id1614695\">Determine whether <a class=\"autogenerated-content\" href=\"#Table_04_08_13\">(Figure)<\/a> could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.<\/p>\n<table id=\"Table_04_08_13\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)\u201d. Reading the columns as ordered pairs, we have the following values: (1, 3), (2, 0.9), (3, 0.27), and (4, 0.081).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<td>3<\/td>\n<td>0.9<\/td>\n<td>0.27<\/td>\n<td>0.081<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1631360\">\n<div id=\"fs-id1631362\">\n<p id=\"fs-id1631364\">A retirement account is opened with an initial deposit of $8,500 and earns[latex]\\,8.12%\\,[\/latex]interest compounded monthly. What will the account be worth in[latex]\\,20\\,[\/latex]years?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1410744\">[latex]$42,888.18[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1676400\">\n<div id=\"fs-id1676402\">\n<p id=\"fs-id1676404\">Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with[latex]\\,7.5%\\,[\/latex]APR, compounded daily, in order to reach her goal in[latex]\\,3\\,[\/latex]years?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1509389\">\n<div id=\"fs-id1509391\">\n<p id=\"fs-id1457722\">Does the equation[latex]\\,y=2.294{e}^{-0.654t}\\,[\/latex]represent continuous growth, continuous decay, or neither? Explain.<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1678302\">continuous decay; the growth rate is negative.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1355174\">\n<div id=\"fs-id1355176\">\n<p id=\"fs-id1355178\">Suppose an investment account is opened with an initial deposit of[latex]\\,\\text{\\$10,500}\\,[\/latex]earning[latex]\\,6.25%\\,[\/latex]interest, compounded continuously. How much will the account be worth after[latex]\\,25\\,[\/latex]years?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1407749\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/73c684c9-6dae-4f32-a1a1-5208b5bf59c2\">Graphs of Exponential Functions<\/a><\/h4>\n<div id=\"fs-id1506677\">\n<div id=\"fs-id1506679\">\n<p id=\"fs-id1506681\">Graph the function[latex]\\,f\\left(x\\right)=3.5{\\left(2\\right)}^{x}.\\,[\/latex]State the domain and range and give the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1594839\">domain: all real numbers; range: all real numbers strictly greater than zero; <em>y<\/em>-intercept: (0, 3.5);<\/p>\n<p><span id=\"fs-id1430386\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141353\/CNX_PreCalc_Figure_04_08_229.jpg\" alt=\"Graph of f(x)=3.5(2^x)\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1440005\">\n<div id=\"fs-id1440007\">\n<p id=\"fs-id1440009\">Graph the function[latex]\\,f\\left(x\\right)=4{\\left(\\frac{1}{8}\\right)}^{x}\\,[\/latex]and its reflection about the <em>y<\/em>-axis on the same axes, and give the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1697407\">\n<div id=\"fs-id1697409\">\n<p id=\"fs-id1697411\">The graph of[latex]\\,f\\left(x\\right)={6.5}^{x}\\,[\/latex]is reflected about the <em>y<\/em>-axis and stretched vertically by a factor of[latex]\\,7.\\,[\/latex]What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1433654\">[latex]g\\left(x\\right)=7{\\left(6.5\\right)}^{-x};\\,[\/latex]<em>y<\/em>-intercept:[latex]\\,\\left(0,\\text{ 7}\\right);\\,[\/latex]Domain: all real numbers; Range: all real numbers greater than[latex]\\,0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1405238\">\n<div id=\"fs-id1405240\">\n<p id=\"fs-id1671097\">The graph below shows transformations of the graph of[latex]\\,f\\left(x\\right)={2}^{x}.\\,[\/latex]What is the equation for the transformation?<\/p>\n<p><span id=\"fs-id1407130\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141405\/CNX_PreCalc_Figure_04_08_231.jpg\" alt=\"Graph of f(x)=2^x\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1586983\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/746b4be7-5dfd-4d01-8293-06ef750e0365\">Logarithmic Functions<\/a><\/h4>\n<div id=\"fs-id1586989\">\n<div id=\"fs-id1350987\">\n<p id=\"fs-id1350989\">Rewrite[latex]\\,{\\mathrm{log}}_{17}\\left(4913\\right)=x\\,[\/latex]as an equivalent exponential equation.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1365015\">[latex]{17}^{x}=4913[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1439372\">\n<div id=\"fs-id1439374\">\n<p id=\"fs-id1380764\">Rewrite[latex]\\,\\mathrm{ln}\\left(s\\right)=t\\,[\/latex]as an equivalent exponential equation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1649693\">\n<div id=\"fs-id1649695\">\n<p id=\"fs-id1649697\">Rewrite[latex]\\,{a}^{-\\,\\frac{2}{5}}=b\\,[\/latex]as an equivalent logarithmic equation.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1569161\">[latex]{\\mathrm{log}}_{a}b=-\\frac{2}{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1431893\">\n<div id=\"fs-id1431895\">\n<p id=\"fs-id1434949\">Rewrite [latex]\\,{e}^{-3.5}=h\\,[\/latex] as an equivalent logarithmic equation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1631001\">\n<div id=\"fs-id1631003\">\n<p id=\"fs-id1631005\">Solve for x if[latex]\\,\\,\\,{\\mathrm{log}}_{64}\\left(x\\right)=\\frac{1}{3}\\,[\/latex]by converting to exponential form.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1705579\">[latex]x={64}^{\\frac{1}{3}}=4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1594722\">\n<div id=\"fs-id1536233\">\n<p id=\"fs-id1536235\">Evaluate[latex]\\,{\\mathrm{log}}_{5}\\left(\\frac{1}{125}\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1658735\">\n<div id=\"fs-id1658737\">\n<p id=\"fs-id1658739\">Evaluate[latex]\\,\\mathrm{log}\\left(\\text{0}\\text{.000001}\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1428405\">[latex]\\mathrm{log}\\left(\\text{0}\\text{.000001}\\right)=-6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1517380\">\n<div id=\"fs-id1685591\">\n<p id=\"fs-id1685593\">Evaluate[latex]\\,\\mathrm{log}\\left(4.005\\right)\\,[\/latex]using a calculator. Round to the nearest thousandth.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1536305\">\n<div id=\"fs-id1536307\">\n<p id=\"fs-id1536309\">Evaluate[latex]\\,\\mathrm{ln}\\left({e}^{-0.8648}\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1434281\">[latex]\\mathrm{ln}\\left({e}^{-0.8648}\\right)=-0.8648[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1405125\">\n<div id=\"fs-id1405127\">\n<p id=\"fs-id1405129\">Evaluate[latex]\\,\\mathrm{ln}\\left(\\sqrt[3]{18}\\right)\\,[\/latex]using a calculator. Round to the nearest thousandth.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1459475\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/44418435-ed46-454a-aba4-cd57f5266654\">Graphs of Logarithmic Functions<\/a><\/h4>\n<div id=\"fs-id1459480\">\n<div id=\"fs-id1459482\">\n<p id=\"fs-id1459484\">Graph the function[latex]\\,g\\left(x\\right)=\\mathrm{log}\\left(7x+21\\right)-4.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141412\/CNX_PreCalc_Figure_04_08_232.jpg\" alt=\"Graph of g(x)=log(7x+21)-4.\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1514032\">\n<div id=\"fs-id1514034\">\n<p id=\"fs-id1514036\">Graph the function[latex]\\,h\\left(x\\right)=2\\mathrm{ln}\\left(9-3x\\right)+1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1365359\">\n<div id=\"fs-id1588507\">\n<p id=\"fs-id1588509\">State the domain, vertical asymptote, and end behavior of the function[latex]\\,g\\left(x\\right)=\\mathrm{ln}\\left(4x+20\\right)-17.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1701036\">Domain:[latex]\\,x>-5;\\,[\/latex]Vertical asymptote:[latex]\\,x=-5;\\,[\/latex]End behavior: as[latex]\\,x\\to -{5}^{+},f\\left(x\\right)\\to -\\infty \\,[\/latex]and as[latex]\\,x\\to \\infty ,f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1807970\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/9d565dd7-7228-45d1-b907-b0e9b7418fb9\">Logarithmic Properties<\/a><\/h4>\n<div id=\"fs-id1865335\">\n<div id=\"fs-id1865338\">\n<p id=\"fs-id1865340\">Rewrite[latex]\\,\\mathrm{ln}\\left(7r\\cdot 11st\\right)\\,[\/latex]in expanded form.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1608999\">\n<div id=\"fs-id1609001\">\n<p id=\"fs-id1609003\">Rewrite[latex]\\,{\\mathrm{log}}_{8}\\left(x\\right)+{\\mathrm{log}}_{8}\\left(5\\right)+{\\mathrm{log}}_{8}\\left(y\\right)+{\\mathrm{log}}_{8}\\left(13\\right)\\,[\/latex]in compact form.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1509129\">[latex]{\\text{log}}_{8}\\left(65xy\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1512031\">\n<div id=\"fs-id1512033\">\n<p id=\"fs-id1428552\">Rewrite[latex]\\,{\\mathrm{log}}_{m}\\left(\\frac{67}{83}\\right)\\,[\/latex]in expanded form.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1294263\">\n<div id=\"fs-id1294265\">\n<p id=\"fs-id1600903\">Rewrite[latex]\\,\\mathrm{ln}\\left(z\\right)-\\mathrm{ln}\\left(x\\right)-\\mathrm{ln}\\left(y\\right)\\,[\/latex]in compact form.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1676007\">[latex]\\mathrm{ln}\\left(\\frac{z}{xy}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1503660\">\n<div id=\"fs-id1503662\">\n<p id=\"fs-id1503664\">Rewrite[latex]\\,\\mathrm{ln}\\left(\\frac{1}{{x}^{5}}\\right)\\,[\/latex]as a product.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1629102\">\n<div id=\"fs-id1629104\">\n<p id=\"fs-id1707011\">Rewrite[latex]\\,-{\\mathrm{log}}_{y}\\left(\\frac{1}{12}\\right)\\,[\/latex]as a single logarithm.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1371100\">[latex]{\\text{log}}_{y}\\left(12\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1433165\">\n<div id=\"fs-id1433167\">\n<p id=\"fs-id1433169\">Use properties of logarithms to expand[latex]\\,\\mathrm{log}\\left(\\frac{{r}^{2}{s}^{11}}{{t}^{14}}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1584205\">\n<div id=\"fs-id1649231\">\n<p id=\"fs-id1649233\">Use properties of logarithms to expand[latex]\\,\\mathrm{ln}\\left(2b\\sqrt{\\frac{b+1}{b-1}}\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1519261\">[latex]\\mathrm{ln}\\left(2\\right)+\\mathrm{ln}\\left(b\\right)+\\frac{\\mathrm{ln}\\left(b+1\\right)-\\mathrm{ln}\\left(b-1\\right)}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1692961\">\n<div id=\"fs-id1443186\">\n<p id=\"fs-id1443188\">Condense the expression[latex]\\,5\\mathrm{ln}\\left(b\\right)+\\mathrm{ln}\\left(c\\right)+\\frac{\\mathrm{ln}\\left(4-a\\right)}{2}\\,[\/latex]to a single logarithm.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1646710\">\n<div id=\"fs-id1646712\">\n<p id=\"fs-id1646714\">Condense the expression[latex]\\,3{\\mathrm{log}}_{7}v+6{\\mathrm{log}}_{7}w-\\frac{{\\mathrm{log}}_{7}u}{3}\\,[\/latex]to a single logarithm.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1419882\">[latex]{\\mathrm{log}}_{7}\\left(\\frac{{v}^{3}{w}^{6}}{\\sqrt[3]{u}}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1582444\">\n<div id=\"fs-id1582446\">\n<p id=\"fs-id1582448\">Rewrite[latex]\\,{\\mathrm{log}}_{3}\\left(12.75\\right)\\,[\/latex]to base[latex]\\,e.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1429357\">\n<div id=\"fs-id1429359\">\n<p id=\"fs-id1429361\">Rewrite[latex]\\,{5}^{12x-17}=125\\,[\/latex]as a logarithm. Then apply the change of base formula to solve for[latex]\\,x\\,[\/latex]using the common log. Round to the nearest thousandth.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1352011\">[latex]x=\\frac{\\frac{\\mathrm{log}\\left(125\\right)}{\\mathrm{log}\\left(5\\right)}+17}{12}=\\frac{5}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1376798\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/c1f8641f-2121-4457-adc2-ef58f23500ce\">Exponential and Logarithmic Equations<\/a><\/h4>\n<div id=\"fs-id1376803\">\n<div id=\"fs-id1376805\">\n<p id=\"fs-id1376808\">Solve[latex]\\,{216}^{3x}\\cdot {216}^{x}={36}^{3x+2}\\,[\/latex]by rewriting each side with a common base.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1421086\">\n<div id=\"fs-id1421088\">\n<p id=\"fs-id1691321\">Solve[latex]\\,\\frac{125}{{\\left(\\frac{1}{625}\\right)}^{-x-3}}={5}^{3}\\,[\/latex]by rewriting each side with a common base.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1376667\">[latex]x=-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1431560\">\n<div id=\"fs-id1431562\">\n<p id=\"fs-id1431564\">Use logarithms to find the exact solution for[latex]\\,7\\cdot {17}^{-9x}-7=49.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1563478\">\n<div id=\"fs-id1434653\">\n<p id=\"fs-id1434655\">Use logarithms to find the exact solution for[latex]\\,3{e}^{6n-2}+1=-60.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1582355\">no solution<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1582361\">\n<div id=\"fs-id1424610\">\n<p id=\"fs-id1424612\">Find the exact solution for[latex]\\,5{e}^{3x}-4=6\\,[\/latex]. If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1608922\">\n<div id=\"fs-id1518514\">\n<p id=\"fs-id1518516\">Find the exact solution for[latex]\\,2{e}^{5x-2}-9=-56.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1671694\">no solution<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1671700\">\n<div id=\"fs-id1602116\">\n<p id=\"fs-id1602118\">Find the exact solution for[latex]\\,{5}^{2x-3}={7}^{x+1}.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1570009\">\n<div id=\"fs-id1570011\">\n<p id=\"fs-id1570013\">Find the exact solution for[latex]\\,{e}^{2x}-{e}^{x}-110=0.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1403522\">[latex]x=\\mathrm{ln}\\left(11\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1698182\">\n<div id=\"fs-id1698185\">\n<p id=\"fs-id1698187\">Use the definition of a logarithm to solve.[latex]\\,-5{\\mathrm{log}}_{7}\\left(10n\\right)=5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1408770\">\n<div id=\"fs-id1408772\">\n<p id=\"fs-id1408774\">47. Use the definition of a logarithm to find the exact solution for[latex]\\,9+6\\mathrm{ln}\\left(a+3\\right)=33.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1609098\">[latex]a={e}^{4}-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1446674\">\n<div id=\"fs-id1446676\">\n<p id=\"fs-id1446678\">Use the one-to-one property of logarithms to find an exact solution for[latex]\\,{\\mathrm{log}}_{8}\\left(7\\right)+{\\mathrm{log}}_{8}\\left(-4x\\right)={\\mathrm{log}}_{8}\\left(5\\right).\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1405530\">\n<div id=\"fs-id1440096\">\n<p id=\"fs-id1440098\">Use the one-to-one property of logarithms to find an exact solution for[latex]\\,\\mathrm{ln}\\left(5\\right)+\\mathrm{ln}\\left(5{x}^{2}-5\\right)=\\mathrm{ln}\\left(56\\right).\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1408805\">[latex]x=\u00b1\\frac{9}{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1395438\">\n<div id=\"fs-id1395440\">\n<p id=\"fs-id1409989\">The formula for measuring sound intensity in decibels[latex]\\,D\\,[\/latex]is defined by the equation[latex]\\,D=10\\mathrm{log}\\left(\\frac{I}{{I}_{0}}\\right),[\/latex] where[latex]\\,I\\,[\/latex]is the intensity of the sound in watts per square meter and[latex]\\,{I}_{0}={10}^{-12}\\,[\/latex]is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of[latex]\\,6.3\\cdot {10}^{-3}\\,[\/latex]watts per square meter?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1376365\">\n<div id=\"fs-id1520052\">\n<p id=\"fs-id1520055\">The population of a city is modeled by the equation[latex]\\,P\\left(t\\right)=256,114{e}^{0.25t}\\,[\/latex]where[latex]\\,t\\,[\/latex]is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1648231\">about[latex]\\,5.45\\,[\/latex]years<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1624824\">\n<div id=\"fs-id1624827\">\n<p id=\"fs-id1624829\">Find the inverse function[latex]\\,{f}^{-1}\\,[\/latex]for the exponential function[latex]\\,f\\left(x\\right)=2\\cdot {e}^{x+1}-5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1520404\">\n<div id=\"fs-id1520406\">\n<p id=\"fs-id1520408\">Find the inverse function[latex]\\,{f}^{-1}\\,[\/latex]for the logarithmic function[latex]\\,f\\left(x\\right)=0.25\\cdot {\\mathrm{log}}_{2}\\left({x}^{3}+1\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1563604\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt[3]{{2}^{4x}-1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1563711\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/feda96a1-a0f3-41ce-9d42-43eef361a909\">Exponential and Logarithmic Models<\/a><\/h4>\n<p id=\"fs-id1563717\">For the following exercises, use this scenario: A doctor prescribes[latex]\\,300\\,[\/latex]milligrams of a therapeutic drug that decays by about[latex]\\,17%\\,[\/latex]each hour.<\/p>\n<div id=\"fs-id1452308\">\n<div id=\"fs-id1393659\">\n<p id=\"fs-id1393661\">To the nearest minute, what is the half-life of the drug?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1393666\">\n<div id=\"fs-id1393668\">\n<p id=\"fs-id1393670\">Write an exponential model representing the amount of the drug remaining in the patient\u2019s system after[latex]\\,t\\,[\/latex]hours. Then use the formula to find the amount of the drug that would remain in the patient\u2019s system after[latex]\\,24\\,[\/latex]hours. Round to the nearest hundredth of a gram.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1671547\">[latex]f\\left(t\\right)=300{\\left(0.83\\right)}^{t};f\\left(24\\right)\\approx 3.43\\text{\u200a}\\text{\u200a}g[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1647206\">For the following exercises, use this scenario: A soup with an internal temperature of[latex]\\,\\text{350\u00b0}\\,[\/latex]Fahrenheit was taken off the stove to cool in a[latex]\\,\\text{71\u00b0F}\\,[\/latex]room. After fifteen minutes, the internal temperature of the soup was[latex]\\,\\text{175\u00b0F}\\text{.}[\/latex]<\/p>\n<div id=\"fs-id1410718\">\n<div id=\"fs-id1410720\">\n<p id=\"fs-id1410722\">Use Newton\u2019s Law of Cooling to write a formula that models this situation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1410729\">\n<div id=\"fs-id1410731\">\n<p id=\"fs-id1422416\">How many minutes will it take the soup to cool to[latex]\\,\\text{85\u00b0F?}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1409376\">about[latex]\\,45\\,[\/latex]minutes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1354791\">For the following exercises, use this scenario: The equation[latex]\\,N\\left(t\\right)=\\frac{1200}{1+199{e}^{-0.625t}}\\,[\/latex]models the number of people in a school who have heard a rumor after[latex]\\,t\\,[\/latex]days.<\/p>\n<div id=\"fs-id1406672\">\n<div id=\"fs-id1406674\">\n<p id=\"fs-id1406676\">How many people started the rumor?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1406889\">\n<div id=\"fs-id1406892\">\n<p id=\"fs-id1406894\">To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1406900\">about[latex]\\,8.5\\,[\/latex]days<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1512099\">\n<div id=\"fs-id1512102\">\n<p id=\"fs-id1512104\">What is the carrying capacity?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1512109\">For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.<\/p>\n<div id=\"fs-id1652654\">\n<div id=\"fs-id1652656\">\n<table id=\"fs-id1652658\" class=\"unnumbered\" summary=\"Two columns and eleven rpws. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 3.05), (2, 4.42), (3, 6.4), (4, 9.28), (5, 13.46), (6, 19.52), (7, 28.3), (8, 41.01), (9, 59.5), and (10, 86.28).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>3.05<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>4.42<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>6.4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>9.28<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>13.46<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>19.52<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>28.3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>41.04<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>59.5<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>86.28<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1403487\">exponential<\/p>\n<p><span id=\"fs-id1406597\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141414\/CNX_PreCalc_Figure_04_08_234.jpg\" alt=\"Graph of the table\u2019s values.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1423620\">\n<div id=\"fs-id1423623\">\n<table id=\"fs-id1423625\" class=\"unnumbered\" summary=\"Two columns and twelve rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (0.5, 18.05), (1, 17), (3, 15.33), (5, 14.55), (7, 14.04), (10, 13.5), (12, 13.22), (13, 13.1), (15, 12.88), (17, 12.69), and (20, 12.45).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>0.5<\/td>\n<td>18.05<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>17<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>15.33<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>14.55<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>14.04<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>13.5<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>13.22<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>13.1<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>12.88<\/td>\n<\/tr>\n<tr>\n<td>17<\/td>\n<td>12.69<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>12.45<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1614793\">\n<div id=\"fs-id1614795\">\n<p id=\"fs-id1614797\">Find a formula for an exponential equation that goes through the points[latex]\\,\\left(-2,100\\right)\\,[\/latex]and[latex]\\,\\left(0,4\\right).\\,[\/latex]Then express the formula as an equivalent equation with base <em>e.<\/em><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1657094\">[latex]y=4{\\left(0.2\\right)}^{x};\\,[\/latex][latex]y=4{e}^{\\text{-1}\\text{.609438}x}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1658476\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/aa3a6479-56b9-4e74-9b72-5817cfe5bf5a\">Fitting Exponential Models to Data<\/a><\/h4>\n<div id=\"fs-id1658482\">\n<div id=\"fs-id1699077\">\n<p id=\"fs-id1699079\">What is the carrying capacity for a population modeled by the logistic equation[latex]\\,P\\left(t\\right)=\\frac{250,000}{1\\,\\,+\\,\\,499{e}^{-0.45t}}?\\,[\/latex]What is the initial population for the model?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1446741\">\n<div id=\"fs-id1446744\">\n<p id=\"fs-id1424342\">The population of a culture of bacteria is modeled by the logistic equation[latex]\\,P\\left(t\\right)=\\frac{14,250}{1\\,\\,+\\,\\,29{e}^{-0.62t}},[\/latex] where[latex]\\,t\\,[\/latex]is in days. To the nearest tenth, how many days will it take the culture to reach[latex]\\,75%\\,[\/latex]of its carrying capacity?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1696243\">about[latex]\\,7.2\\,[\/latex]days<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1562633\">For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.<\/p>\n<div id=\"fs-id1597899\">\n<div id=\"fs-id1597901\">\n<table id=\"fs-id1597902\" class=\"unnumbered\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 409.4), (2, 260.7), (3, 170.4), (4, 110.6), (5, 74), (6, 44.7), (7, 32.4), (8, 19.5), (9, 12.7), and (10, 8.1).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>409.4<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>260.7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>170.4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>110.6<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>74<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>44.7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>32.4<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>19.5<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>12.7<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>8.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1761662\">\n<div id=\"fs-id1761665\">\n<table id=\"fs-id1761667\" class=\"unnumbered\" summary=\"Two rows and twelve columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)\u201d. Reading the columns as ordered pairs, we have the following values: (0.15, 36.21), (0.25, 28.88), (0.5, 24.39), (0.75, 18.28), (1, 16.5), (1.5, 12.99), (2, 9.91), (2.25, 8.57), (2.75, 7.23), (3, 5.99), and (3.5, 4.81).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>0.15<\/td>\n<td>36.21<\/td>\n<\/tr>\n<tr>\n<td>0.25<\/td>\n<td>28.88<\/td>\n<\/tr>\n<tr>\n<td>0.5<\/td>\n<td>24.39<\/td>\n<\/tr>\n<tr>\n<td>0.75<\/td>\n<td>18.28<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>16.5<\/td>\n<\/tr>\n<tr>\n<td>1.5<\/td>\n<td>12.99<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>9.91<\/td>\n<\/tr>\n<tr>\n<td>2.25<\/td>\n<td>8.57<\/td>\n<\/tr>\n<tr>\n<td>2.75<\/td>\n<td>7.23<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>5.99<\/td>\n<\/tr>\n<tr>\n<td>3.5<\/td>\n<td>4.81<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1705002\">logarithmic;[latex]\\,y=16.68718-9.71860\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1421423\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141434\/CNX_PreCalc_Figure_04_08_237.jpg\" alt=\"Graph of the table\u2019s values.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1408475\">\n<div id=\"fs-id1408477\">\n<table id=\"fs-id1408479\" class=\"unnumbered\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (0, 9), (2, 22.6), (4, 44.2), (5, 62.1), (7, 96.9), (8, 113.4), (10, 133.4), (11, 137.6), (15, 148.4), and (17, 149.3).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>22.6<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>44.2<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>62.1<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>96.9<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>113.4<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>133.4<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>137.6<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>148.4<\/td>\n<\/tr>\n<tr>\n<td>17<\/td>\n<td>149.3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1513725\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<div id=\"fs-id1513728\">\n<div id=\"fs-id1513730\">\n<p id=\"fs-id1513732\">The population of a pod of bottlenose dolphins is modeled by the function[latex]\\,A\\left(t\\right)=8{\\left(1.17\\right)}^{t},[\/latex] where[latex]\\,t\\,[\/latex]is given in years. To the nearest whole number, what will the pod population be after[latex]\\,3\\,[\/latex]years?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1597336\">About[latex]\\,13\\,[\/latex]dolphins.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1664103\">\n<div id=\"fs-id1664105\">\n<p id=\"fs-id1664107\">Find an exponential equation that passes through the points[latex]\\,\\text{(0, 4)}\\,[\/latex]and[latex]\\,\\text{(2, 9)}\\text{.}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1530968\">\n<div id=\"fs-id1530970\">\n<p id=\"fs-id1530972\">Drew wants to save $2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with[latex]\\,6.25%\\,[\/latex]APR, compounding daily, in order to reach his goal in[latex]\\,4\\,[\/latex]years?<\/p>\n<\/div>\n<div id=\"fs-id723927\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id723929\">[latex]$1,947[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1457825\">\n<div id=\"fs-id1457827\">\n<p id=\"fs-id1457829\">An investment account was opened with an initial deposit of $9,600 and earns[latex]\\,7.4%\\,[\/latex]interest, compounded continuously. How much will the account be worth after[latex]\\,15\\,[\/latex]years?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1375232\">\n<div id=\"fs-id1375234\">\n<p id=\"fs-id1375236\">Graph the function[latex]\\,f\\left(x\\right)=5{\\left(0.5\\right)}^{-x}\\,[\/latex]and its reflection across the <em>y<\/em>-axis on the same axes, and give the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1588458\"><em>y<\/em>-intercept:[latex]\\,\\left(0,\\text{ 5}\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1357638\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141444\/CNX_PreCalc_Figure_04_08_239.jpg\" alt=\"Graph of f(-x)=5(0.5)^-x in blue and f(x)=5(0.5)^x in orange.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1691088\">\n<div id=\"fs-id1691090\">\n<p id=\"fs-id1691092\">The graph shows transformations of the graph of[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.\\,[\/latex]What is the equation for the transformation?<\/p>\n<p><span id=\"fs-id1586380\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141451\/CNX_PreCalc_Figure_04_08_240.jpg\" alt=\"Graph of f(x)= (1\/2)^x.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1407487\">\n<div id=\"fs-id1407489\">\n<p id=\"fs-id1407491\">Rewrite[latex]\\,{\\mathrm{log}}_{8.5}\\left(614.125\\right)=a\\,[\/latex]as an equivalent exponential equation.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1653046\">[latex]{8.5}^{a}=614.125[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1487536\">\n<div id=\"fs-id1487538\">\n<p id=\"fs-id1338247\">Rewrite[latex]\\,{e}^{\\frac{1}{2}}=m\\,[\/latex]as an equivalent logarithmic equation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1393960\">\n<div id=\"fs-id1393962\">\n<p id=\"fs-id1393964\">Solve for[latex]\\,x\\,[\/latex]by converting the logarithmic equation[latex]\\,lo{g}_{\\frac{1}{7}}\\left(x\\right)=2\\,[\/latex]to exponential form.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1361043\">[latex]x={\\left(\\frac{1}{7}\\right)}^{2}=\\frac{1}{49}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1630279\">\n<div id=\"fs-id1630281\">\n<p id=\"fs-id1630283\">Evaluate[latex]\\,\\mathrm{log}\\left(\\text{10,000,000}\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1430674\">\n<div id=\"fs-id1423430\">\n<p id=\"fs-id1423432\">Evaluate[latex]\\,\\mathrm{ln}\\left(0.716\\right)\\,[\/latex]using a calculator. Round to the nearest thousandth.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1460620\">[latex]\\mathrm{ln}\\left(0.716\\right)\\approx -0.334[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1405885\">\n<div id=\"fs-id1405888\">\n<p id=\"fs-id1405890\">Graph the function[latex]\\,g\\left(x\\right)=\\mathrm{log}\\left(12-6x\\right)+3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1651954\">\n<div id=\"fs-id1651956\">\n<p id=\"fs-id1651958\">State the domain, vertical asymptote, and end behavior of the function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{5}\\left(39-13x\\right)+7.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1678313\">Domain:[latex]\\,x<3;\\,[\/latex]Vertical asymptote:[latex]\\,x=3;\\,[\/latex]End behavior:[latex]\\,x\\to {3}^{-},f\\left(x\\right)\\to -\\infty \\,[\/latex]and[latex]\\,x\\to -\\infty ,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1423972\">\n<div id=\"fs-id1423974\">\n<p id=\"fs-id1423976\">Rewrite[latex]\\,\\mathrm{log}\\left(17a\\cdot 2b\\right)\\,[\/latex]as a sum.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1508077\">\n<div id=\"fs-id1430560\">\n<p id=\"fs-id1430562\">Rewrite[latex]\\,{\\mathrm{log}}_{t}\\left(96\\right)-{\\mathrm{log}}_{t}\\left(8\\right)\\,[\/latex]in compact form.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1406644\">[latex]{\\mathrm{log}}_{t}\\left(12\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1405262\">\n<div id=\"fs-id1405264\">\n<p id=\"fs-id1405267\">Rewrite[latex]\\,{\\mathrm{log}}_{8}\\left({a}^{\\frac{1}{b}}\\right)\\,[\/latex]as a product.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1690738\">\n<div id=\"fs-id1690740\">\n<p id=\"fs-id1690742\">Use properties of logarithm to expand[latex]\\,\\mathrm{ln}\\left({y}^{3}{z}^{2}\\cdot \\sqrt[3]{x-4}\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1439808\">[latex]3\\,\\,\\mathrm{ln}\\left(y\\right)+2\\mathrm{ln}\\left(z\\right)+\\frac{\\mathrm{ln}\\left(x-4\\right)}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1660668\">\n<div id=\"fs-id1660670\">\n<p id=\"fs-id1660672\">Condense the expression[latex]\\,4\\mathrm{ln}\\left(c\\right)+\\mathrm{ln}\\left(d\\right)+\\frac{\\mathrm{ln}\\left(a\\right)}{3}+\\frac{\\mathrm{ln}\\left(b+3\\right)}{3}\\,[\/latex]to a single logarithm.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1417371\">\n<div id=\"fs-id1417373\">\n<p id=\"fs-id1528728\">Rewrite[latex]\\,{16}^{3x-5}=1000\\,[\/latex]as a logarithm. Then apply the change of base formula to solve for[latex]\\,x\\,[\/latex]using the natural log. Round to the nearest thousandth.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1647240\">[latex]x=\\frac{\\frac{\\mathrm{ln}\\left(1000\\right)}{\\mathrm{ln}\\left(16\\right)}+5}{3}\\approx 2.497[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1395132\">\n<div id=\"fs-id1395134\">\n<p id=\"fs-id1395136\">Solve[latex]\\,{\\left(\\frac{1}{81}\\right)}^{x}\\cdot \\frac{1}{243}={\\left(\\frac{1}{9}\\right)}^{-3x-1}\\,[\/latex]by rewriting each side with a common base.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1646539\">\n<div id=\"fs-id1646541\">\n<p id=\"fs-id1646543\">Use logarithms to find the exact solution for[latex]\\,-9{e}^{10a-8}-5=-41[\/latex]. If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1700907\">[latex]a=\\frac{\\mathrm{ln}\\left(4\\right)+8}{10}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1365542\">\n<div id=\"fs-id1365544\">\n<p id=\"fs-id1365546\">Find the exact solution for[latex]\\,10{e}^{4x+2}+5=56.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1522787\">\n<div id=\"fs-id1522789\">\n<p id=\"fs-id1522792\">Find the exact solution for[latex]\\,-5{e}^{-4x-1}-4=64.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1402901\">no solution<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1402906\">\n<div id=\"fs-id1402909\">\n<p id=\"fs-id1402911\">Find the exact solution for[latex]\\,{2}^{x-3}={6}^{2x-1}.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1407851\">\n<div id=\"fs-id1407853\">\n<p id=\"fs-id1407855\">Find the exact solution for[latex]\\,{e}^{2x}-{e}^{x}-72=0.\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1434398\">[latex]x=\\mathrm{ln}\\left(9\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1435523\">\n<div id=\"fs-id1435525\">\n<p id=\"fs-id1435527\">Use the definition of a logarithm to find the exact solution for[latex]\\,4\\mathrm{log}\\left(2n\\right)-7=-11[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1589316\">\n<div id=\"fs-id1589318\">\n<p id=\"fs-id1589320\">Use the one-to-one property of logarithms to find an exact solution for[latex]\\,\\mathrm{log}\\left(4{x}^{2}-10\\right)+\\mathrm{log}\\left(3\\right)=\\mathrm{log}\\left(51\\right)\\,[\/latex]If there is no solution, write <em>no solution<\/em>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1435205\">[latex]x=\u00b1\\frac{3\\sqrt{3}}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1408912\">\n<div id=\"fs-id1408914\">\n<p id=\"fs-id1522030\">The formula for measuring sound intensity in decibels[latex]\\,D\\,[\/latex]is defined by the equation[latex]\\,D=10\\mathrm{log}\\left(\\frac{I}{{I}_{0}}\\right),[\/latex]where[latex]\\,I\\,[\/latex]is the intensity of the sound in watts per square meter and[latex]\\,{I}_{0}={10}^{-12}\\,[\/latex]is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of[latex]\\,4.7\\cdot {10}^{-1}\\,[\/latex]watts per square meter?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1457675\">\n<div id=\"fs-id1457677\">\n<p id=\"fs-id1457680\">A radiation safety officer is working with[latex]\\,112\\,[\/latex]grams of a radioactive substance. After[latex]\\,17\\,[\/latex]days, the sample has decayed to[latex]\\,80\\,[\/latex]grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]f\\left(t\\right)=112{e}^{-.019792t};[\/latex] half-life: about[latex]\\,35\\,[\/latex] days<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1432306\">\n<div id=\"fs-id1432309\">\n<p id=\"fs-id1432311\">Write the formula found in the previous exercise as an equivalent equation with base[latex]\\,e.\\,[\/latex]Express the exponent to five significant digits.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1405159\">\n<div id=\"fs-id1405161\">\n<p id=\"fs-id1405163\">A bottle of soda with a temperature of[latex]\\,\\text{71\u00b0}\\,[\/latex]Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of[latex]\\,\\text{35\u00b0 F}\\text{.}\\,[\/latex]After ten minutes, the internal temperature of the soda was[latex]\\,\\text{63\u00b0 F}\\text{.}\\,[\/latex]Use Newton\u2019s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1700652\">[latex]T\\left(t\\right)=36{e}^{-0.025131t}+35;T\\left(60\\right)\\approx {43}^{\\text{o}}\\text{F}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1487596\">\n<div id=\"fs-id1504591\">\n<p id=\"fs-id1504593\">The population of a wildlife habitat is modeled by the equation[latex]\\,P\\left(t\\right)=\\frac{360}{1+6.2{e}^{-0.35t}},[\/latex] where[latex]\\,t\\,[\/latex]is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1517136\">\n<div id=\"fs-id1517138\">\n<p>Enter the data from <a class=\"autogenerated-content\" href=\"#Table_04_08_14\">(Figure)<\/a> into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.<\/p>\n<table id=\"Table_04_08_14\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 3), (2, 8.55), (3, 11.79), (4, 14.09), (5, 15.88), (6, 17.33), (7, 18.57), (8, 19.64), (9, 20.58), and (10, 21.42).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>8.55<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>11.79<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>14.09<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>15.88<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>17.33<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>18.57<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>19.64<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>20.58<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>21.42<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1665023\">logarithmic<\/p>\n<p><span id=\"fs-id1665029\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141505\/CNX_PreCalc_Figure_04_08_242.jpg\" alt=\"Graph of the table\u2019s values.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1665041\">\n<div id=\"fs-id1665043\">\n<p id=\"fs-id1678197\">The population of a lake of fish is modeled by the logistic equation[latex]\\,P\\left(t\\right)=\\frac{16,120}{1+25{e}^{-0.75t}},[\/latex] where[latex]\\,t\\,[\/latex]is time in years. To the nearest hundredth, how many years will it take the lake to reach[latex]\\,80%\\,[\/latex]of its carrying capacity?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1705179\">For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.<\/p>\n<div id=\"fs-id1705185\">\n<div id=\"fs-id1600290\">\n<table id=\"fs-id1600292\" class=\"unnumbered\" summary=\"Two columns and eleven rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (1, 20), (2, 21.6), (3, 29.2), (4, 36.4), (5, 46.6), (6, 55.7), (7, 72.6), (8, 87.1), (9, 107.2), and (10, 138.1).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>20<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>21.6<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>29.2<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>36.4<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>46.6<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>55.7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>72.6<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>87.1<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>107.2<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>138.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1465082\">exponential;[latex]\\,y=15.10062{\\left(1.24621\\right)}^{x}[\/latex]<\/p>\n<p><span id=\"fs-id1512245\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141508\/CNX_PreCalc_Figure_04_08_243.jpg\" alt=\"Graph of the table\u2019s values.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1300030\">\n<div id=\"fs-id1300032\">\n<table id=\"fs-id1300035\" class=\"unnumbered\" summary=\"Two columns and twelve rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (3, 13.98), (4, 17.84), (5, 20.01), (6, 22.7), (7, 24.1), (8, 26.15), (9, 27.37), (10, 28.38), (11, 29.97), (12, 31.07), and (13, 31.43).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>13.98<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>17.84<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>20.01<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>22.7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>24.1<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>26.15<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>27.37<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>28.38<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>29.97<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>31.07<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>31.43<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1676036\">\n<div id=\"fs-id1676038\">\n<table id=\"fs-id1676040\" class=\"unnumbered\" summary=\"Two columns and twelve rows. The first column is labeled, \u201cx\u201d, and the second column is labeled, \u201cf(x)\u201d. Reading the rows as ordered pairs, we have the following values: (0, 2.2), (0.5, 2.9), (1, 3.9), (1.5,4.8), (2, 6.4), (3, 9.3), (4, 12.3), (5, 15), (6, 16.2), (7, 17.3), and (8, 17.9).\">\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td><strong><em>f(x)<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>2.2<\/td>\n<\/tr>\n<tr>\n<td>0.5<\/td>\n<td>2.9<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>3.9<\/td>\n<\/tr>\n<tr>\n<td>1.5<\/td>\n<td>4.8<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>6.4<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>9.3<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>12.3<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>15<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>16.2<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>17.3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>17.9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1503926\">logistic;[latex]\\,y=\\frac{18.41659}{1+7.54644{e}^{-0.68375x}}[\/latex]<\/p>\n<p><span id=\"fs-id1406292\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141524\/CNX_PreCalc_Figure_04_08_245.jpg\" alt=\"Graph of the table\u2019s values.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-112-1\">\u2022Source: Indiana University Center for Studies of Law in Action, 2007 <a href=\"#return-footnote-112-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-112-2\">\u2022Source: Center for Disease Control and Prevention, 2013 <a href=\"#return-footnote-112-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-112-3\">\u2022Source: The World Bank, 2013 <a href=\"#return-footnote-112-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":291,"menu_order":9,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-112","chapter","type-chapter","status-publish","hentry"],"part":95,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/112","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/112\/revisions"}],"predecessor-version":[{"id":113,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/112\/revisions\/113"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/95"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/112\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=112"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=112"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=112"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}