{"id":102,"date":"2019-08-20T17:02:30","date_gmt":"2019-08-20T21:02:30","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/logarithmic-functions\/"},"modified":"2022-06-01T10:39:28","modified_gmt":"2022-06-01T14:39:28","slug":"logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/logarithmic-functions\/","title":{"raw":"Logarithmic Functions","rendered":"Logarithmic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Convert from logarithmic to exponential form.<\/li>\n \t<li>Convert from exponential to logarithmic form.<\/li>\n \t<li>Evaluate logarithms.<\/li>\n \t<li>Use common logarithms.<\/li>\n \t<li>Use natural logarithms.<\/li>\n<\/ul>\n<\/div>\n<div id=\"CNX_Precalc_Figure_04_03_001\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140521\/CNX_Precalc_Figure_04_03_001.jpg\" alt=\"Photo of the aftermath of the earthquake in Japan with a focus on the Japanese flag.\" width=\"488\" height=\"325\"> <strong>Figure 1. <\/strong>Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)[\/caption]\n\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\n<p id=\"fs-id1165137557013\">In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes[footnote]<a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary<\/a>. Accessed 3\/4\/2013.[\/footnote] . One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,[footnote]<a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary<\/a>. Accessed 3\/4\/2013.[\/footnote] like those shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_03_001\">(Figure)<\/a>. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale[footnote]http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/. Accessed 3\/4\/2013.[\/footnote] whereas the Japanese earthquake registered a 9.0.[footnote]<a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/<\/a>. Accessed 3\/4\/2013.[\/footnote]<\/p>\n<p id=\"fs-id1165137760714\">The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is [latex]{10}^{8-4}={10}^{4}=10,000[\/latex] times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.<\/p>\n\n<div id=\"fs-id1165137644550\" class=\"bc-section section\">\n<h3>Converting from Logarithmic to Exponential Form<\/h3>\n<p id=\"fs-id1165135192781\">In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is[latex]\\,{10}^{x}=500,[\/latex] where[latex]\\,x\\,[\/latex]represents the difference in magnitudes on the <span class=\"no-emphasis\">Richter Scale<\/span>. How would we solve for[latex]\\,x?[\/latex]<\/p>\n<p id=\"fs-id1165135160312\">We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve[latex]\\,{10}^{x}=500.\\,[\/latex]We know that[latex]\\,{10}^{2}=100\\,[\/latex]and[latex]\\,{10}^{3}=1000,[\/latex] so it is clear that[latex]\\,x\\,[\/latex]must be some value between 2 and 3, since[latex]\\,y={10}^{x}\\,[\/latex]is increasing. We can examine a graph, as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_03_002\">(Figure)<\/a><strong>,<\/strong> to better estimate the solution.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_03_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\/19140524\/CNX_Precalc_Figure_04_03_002.jpg\" alt=\"Graph of the intersections of the equations y=10^x and y=500.\" width=\"487\" height=\"477\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137662989\">Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_03_002\">(Figure)<\/a> passes the horizontal line test. The exponential function[latex]\\,y={b}^{x}\\,[\/latex]is <span class=\"no-emphasis\">one-to-one<\/span>, so its inverse,[latex]\\,x={b}^{y}\\,[\/latex]is also a function. As is the case with all inverse functions, we simply interchange[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]and solve for[latex]\\,y\\,[\/latex]to find the inverse function. To represent[latex]\\,y\\,[\/latex]as a function of[latex]\\,x,[\/latex] we use a logarithmic function of the form[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right).\\,[\/latex]The base[latex]\\,b\\,[\/latex]<strong>logarithm<\/strong> of a number is the exponent by which we must raise[latex]\\,b\\,[\/latex]to get that number.<\/p>\n<p id=\"fs-id1165137404844\">We read a logarithmic expression as, \u201cThe logarithm with base[latex]\\,b\\,[\/latex]of[latex]\\,x\\,[\/latex]is equal to[latex]\\,y,[\/latex]\u201d or, simplified, \u201clog base[latex]\\,b\\,[\/latex]of[latex]\\,x\\,[\/latex]is[latex]\\,y.[\/latex]\u201d We can also say, \u201c[latex]b\\,[\/latex]raised to the power of[latex]\\,y\\,[\/latex]is[latex]\\,x,[\/latex]\u201d because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since[latex]\\,{2}^{5}=32,[\/latex] we can write[latex]\\,{\\mathrm{log}}_{2}32=5.\\,[\/latex]We read this as \u201clog base 2 of 32 is 5.\u201d<\/p>\n<p id=\"fs-id1165137597501\">We can express the relationship between logarithmic form and its corresponding exponential form as follows:<\/p>\n\n<div id=\"eip-604\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y\u21d4{b}^{y}=x,\\text{}b&gt;0,b\\ne 1[\/latex]<\/div>\n<p id=\"fs-id1165137678993\">Note that the base[latex]\\,b\\,[\/latex]is always positive.<\/p>\n<span id=\"fs-id1165137696233\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140526\/CNX_Precalc_Figure_04_03_004.jpg\" alt=\"\"><\/span>\n\nBecause logarithm is a function, it is most correctly written as[latex]\\,{\\mathrm{log}}_{b}\\left(x\\right),[\/latex] using parentheses to denote function evaluation, just as we would with[latex]\\,f\\left(x\\right).\\,[\/latex]However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as[latex]\\,{\\mathrm{log}}_{b}x.\\,[\/latex]Note that many calculators require parentheses around the[latex]\\,x.[\/latex]\n<p id=\"fs-id1165137827516\">We can illustrate the notation of logarithms as follows:<\/p>\n<span id=\"fs-id1165137771679\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140532\/CNX_Precalc_Figure_04_03_003.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137575165\">Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]and[latex]\\,y={b}^{x}\\,[\/latex]are inverse functions.<\/p>\n\n<div id=\"fs-id1165137472937\" class=\"textbox key-takeaways\">\n<h3>Definition of the Logarithmic Function<\/h3>\n<p id=\"fs-id1165137704597\">A logarithm base[latex]\\,b\\,[\/latex]of a positive number[latex]\\,x\\,[\/latex]satisfies the following definition.<\/p>\n<p id=\"fs-id1165137584967\">For[latex]\\,x&gt;0,b&gt;0,b\\ne 1,[\/latex]<\/p>\n\n<div id=\"fs-id1165137433829\">[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\text{ is equivalent to }{b}^{y}=x[\/latex]<\/div>\n<p id=\"fs-id1165137893373\">where,<\/p>\n\n<ul id=\"fs-id1165135530561\">\n \t<li>we read[latex]\\,{\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]as, \u201cthe logarithm with base[latex]\\,b\\,[\/latex]of[latex]\\,x[\/latex]\u201d or the \u201clog base[latex]\\,b\\,[\/latex]of[latex]\\,x.\"[\/latex]<\/li>\n \t<li>the logarithm[latex]\\,y\\,[\/latex]is the exponent to which[latex]\\,b\\,[\/latex]must be raised to get[latex]\\,x.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137547773\">Also, since the logarithmic and exponential functions switch the[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,<\/p>\n\n<ul id=\"fs-id1165137643167\">\n \t<li>the domain of the logarithm function with base[latex]\\,b \\text{is} \\left(0,\\infty \\right).[\/latex]<\/li>\n \t<li>the range of the logarithm function with base[latex]\\,b \\text{is} \\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137677696\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1549475\"><strong>Can we take the logarithm of a negative number?<\/strong><\/p>\n<p id=\"fs-id1165137653864\"><em>No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1165137874700\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137806301\"><strong>Given an equation in logarithmic form<\/strong>[latex]\\,{\\mathrm{log}}_{b}\\left(x\\right)=y,[\/latex]<strong> convert it to exponential form.<\/strong><\/p>\n\n<ol id=\"fs-id1165137641669\" type=\"1\">\n \t<li>Examine the equation[latex]\\,y={\\mathrm{log}}_{b}x\\,[\/latex]and identify[latex]\\,b,y,\\text{and}x.[\/latex]<\/li>\n \t<li>Rewrite[latex]\\,{\\mathrm{log}}_{b}x=y\\,[\/latex]as[latex]\\,{b}^{y}=x.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135570363\">\n<div id=\"fs-id1165137557855\">\n<h3>Converting from Logarithmic Form to Exponential Form<\/h3>\n<p id=\"fs-id1165137580570\">Write the following logarithmic equations in exponential form.<\/p>\n\n<ol id=\"fs-id1165137705346\" type=\"a\">\n \t<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]<\/li>\n \t<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135613330\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135613330\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135613330\"]\n<p id=\"fs-id1165137408172\">First, identify the values of[latex]\\,b,y,\\text{and}x.\\,[\/latex]Then, write the equation in the form[latex]\\,{b}^{y}=x.[\/latex]<\/p>\n\n<ol id=\"fs-id1165137705659\" type=\"a\">\n \t<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]\n<p id=\"fs-id1165137602796\">Here,[latex]\\,b=6,y=\\frac{1}{2},\\text{and} x=\\sqrt{6.}\\,[\/latex]Therefore, the equation[latex]\\,{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}\\,[\/latex]is equivalent to[latex]\\,{6}^{\\frac{1}{2}}=\\sqrt{6}.[\/latex]<\/p>\n<\/li>\n \t<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]\n<p id=\"fs-id1165137698078\">Here,[latex]\\,b=3,y=2,\\text{and} x=9.\\,[\/latex]Therefore, the equation[latex]\\,{\\mathrm{log}}_{3}\\left(9\\right)=2\\,[\/latex]is equivalent to[latex]\\,{3}^{2}=9.\\,[\/latex]<\/p>\n<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640140\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165135208926\">\n<p id=\"fs-id1165137418681\">Write the following logarithmic equations in exponential form.<\/p>\n\n<ol id=\"fs-id1165137772342\" type=\"a\">\n \t<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex]<\/li>\n \t<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135195688\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135195688\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135195688\"]\n<ol id=\"fs-id1165137414337\" type=\"a\">\n \t<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6\\,[\/latex]is equivalent to[latex]\\,{10}^{6}=1,000,000[\/latex]<\/li>\n \t<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2\\,[\/latex]is equivalent to[latex]\\,{5}^{2}=25[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137585244\" class=\"bc-section section\">\n<h3>Converting from Exponential to Logarithmic Form<\/h3>\n<p id=\"fs-id1165137933968\">To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base[latex]\\,b,[\/latex]exponent[latex]\\,x,[\/latex]and output[latex]\\,y.\\,[\/latex]Then we write[latex]\\,x={\\mathrm{log}}_{b}\\left(y\\right).[\/latex]<\/p>\n\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135168111\">\n<div id=\"fs-id1165137727912\">\n<h3>Converting from Exponential Form to Logarithmic Form<\/h3>\n<p id=\"fs-id1165137804412\">Write the following exponential equations in logarithmic form.<\/p>\n\n<ol id=\"fs-id1165135192287\" type=\"a\">\n \t<li>[latex]{2}^{3}=8[\/latex]<\/li>\n \t<li>[latex]{5}^{2}=25[\/latex]<\/li>\n \t<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137702205\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137702205\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137702205\"]\n<p id=\"fs-id1165137474116\">First, identify the values of[latex]\\,b,y,\\text{and}x.\\,[\/latex]Then, write the equation in the form[latex]\\,x={\\mathrm{log}}_{b}\\left(y\\right).[\/latex]<\/p>\n\n<ol id=\"fs-id1165137573458\" type=\"a\">\n \t<li>[latex]{2}^{3}=8[\/latex]\n<p id=\"fs-id1165137466396\">Here,[latex]\\,b=2,[\/latex][latex]\\,x=3,[\/latex]and[latex]\\,y=8.\\,[\/latex]Therefore, the equation[latex]\\,{2}^{3}=8\\,[\/latex]is equivalent to[latex]\\,{\\mathrm{log}}_{2}\\left(8\\right)=3.[\/latex]<\/p>\n<\/li>\n \t<li>[latex]{5}^{2}=25[\/latex]\n<p id=\"fs-id1165135193035\">Here,[latex]\\,b=5,[\/latex][latex]\\,x=2,[\/latex]and[latex]\\,y=25.\\,[\/latex]Therefore, the equation[latex]\\,{5}^{2}=25\\,[\/latex]is equivalent to[latex]\\,{\\mathrm{log}}_{5}\\left(25\\right)=2.[\/latex]<\/p>\n<\/li>\n \t<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]\n<p id=\"fs-id1165135187822\">Here,[latex]\\,b=10,[\/latex][latex]\\,x=-4,[\/latex]and[latex]\\,y=\\frac{1}{10,000}.\\,[\/latex]Therefore, the equation[latex]\\,{10}^{-4}=\\frac{1}{10,000}\\,[\/latex]is equivalent to[latex]\\,{\\text{log}}_{10}\\left(\\frac{1}{10,000}\\right)=-4.[\/latex]<\/p>\n<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438165\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165135190969\">\n<p id=\"fs-id1165137566762\">Write the following exponential equations in logarithmic form.<\/p>\n\n<ol id=\"fs-id1165137771963\" type=\"a\">\n \t<li>[latex]{3}^{2}=9[\/latex]<\/li>\n \t<li>[latex]{5}^{3}=125[\/latex]<\/li>\n \t<li>[latex]{2}^{-1}=\\frac{1}{2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165134065138\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134065138\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134065138\"]\n<ol id=\"fs-id1165137469846\" type=\"a\">\n \t<li>[latex]{3}^{2}=9\\,[\/latex]is equivalent to[latex]\\,{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]<\/li>\n \t<li>[latex]{5}^{3}=125\\,[\/latex]is equivalent to[latex]\\,{\\mathrm{log}}_{5}\\left(125\\right)=3[\/latex]<\/li>\n \t<li>[latex]{2}^{-1}=\\frac{1}{2}\\,[\/latex]is equivalent to[latex]\\,{\\text{log}}_{2}\\left(\\frac{1}{2}\\right)=-1[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137530906\" class=\"bc-section section\">\n<h3>Evaluating Logarithms<\/h3>\n<p id=\"fs-id1165137422589\">Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider[latex]\\,{\\mathrm{log}}_{2}8.\\,[\/latex]We ask, \u201cTo what exponent must [latex]\\,2\\,[\/latex] be raised in order to get 8?\u201d Because we already know[latex]\\,{2}^{3}=8,[\/latex] it follows that[latex]\\,{\\mathrm{log}}_{2}8=3.[\/latex]<\/p>\n<p id=\"fs-id1165137733822\">Now consider solving[latex]\\,{\\mathrm{log}}_{7}49\\,[\/latex]and[latex]\\,{\\mathrm{log}}_{3}27\\,[\/latex]mentally.<\/p>\n\n<ul id=\"fs-id1165137937690\">\n \t<li>We ask, \u201cTo what exponent must 7 be raised in order to get 49?\u201d We know[latex]\\,{7}^{2}=49.\\,[\/latex]Therefore,[latex]\\,{\\mathrm{log}}_{7}49=2[\/latex]<\/li>\n \t<li>We ask, \u201cTo what exponent must 3 be raised in order to get 27?\u201d We know[latex]\\,{3}^{3}=27.\\,[\/latex]Therefore,[latex]\\,{\\mathrm{log}}_{3}27=3[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137456358\">Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate[latex]\\,{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}\\,[\/latex]mentally.<\/p>\n\n<ul id=\"fs-id1165137584208\">\n \t<li>We ask, \u201cTo what exponent must[latex]\\,\\frac{2}{3}\\,[\/latex]be raised in order to get[latex]\\,\\frac{4}{9}?\\,[\/latex]\u201d We know[latex]\\,{2}^{2}=4\\,[\/latex]and[latex]\\,{3}^{2}=9,[\/latex]so[latex]\\,{\\left(\\frac{2}{3}\\right)}^{2}=\\frac{4}{9}.\\,[\/latex]Therefore,[latex]\\,{\\mathrm{log}}_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2.[\/latex]<\/li>\n<\/ul>\n<div id=\"fs-id1165137455840\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137453770\"><strong>Given a logarithm of the form[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right),[\/latex]evaluate it mentally.<\/strong><\/p>\n\n<ol id=\"fs-id1165134079724\" type=\"1\">\n \t<li>Rewrite the argument[latex]\\,x\\,[\/latex]as a power of[latex]\\,b:\\,[\/latex][latex]{b}^{y}=x.\\,[\/latex]<\/li>\n \t<li>Use previous knowledge of powers of[latex]\\,b\\,[\/latex]identify[latex]\\,y\\,[\/latex]by asking, \u201cTo what exponent should[latex]\\,b\\,[\/latex]be raised in order to get[latex]\\,x?[\/latex]\u201d<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137732842\">\n<div id=\"fs-id1165135296345\">\n<h3>Solving Logarithms Mentally<\/h3>\n<p id=\"fs-id1165135393440\">Solve[latex]\\,y={\\mathrm{log}}_{4}\\left(64\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137852123\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137852123\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137852123\"]\n<p id=\"fs-id1165137611276\">First we rewrite the logarithm in exponential form:[latex]\\,{4}^{y}=64.\\,[\/latex]Next, we ask, \u201cTo what exponent must 4 be raised in order to get 64?\u201d<\/p>\nWe know\n<div id=\"eip-id1165134583995\" class=\"unnumbered\">[latex]{4}^{3}=64[\/latex]<\/div>\n<p id=\"fs-id1165137619013\">Therefore,<\/p>\n\n<div id=\"eip-id1165135606935\" class=\"unnumbered\">[latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731430\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_03\">\n<div id=\"fs-id1165137704553\">\n<p id=\"fs-id1165137745041\">Solve[latex]\\,y={\\mathrm{log}}_{121}\\left(11\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137693554\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137693554\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137693554\"]\n<p id=\"fs-id1165137639199\">[latex]{\\mathrm{log}}_{121}\\left(11\\right)=\\frac{1}{2}\\,[\/latex](recalling that[latex]\\,\\sqrt{121}={\\left(121\\right)}^{\\frac{1}{2}}=11[\/latex])<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137663658\">\n<div id=\"fs-id1165137680390\">\n<h3>Evaluating the Logarithm of a Reciprocal<\/h3>\n<p id=\"fs-id1165137938805\">Evaluate[latex]\\,y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135526087\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135526087\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135526087\"]\n<p id=\"fs-id1165137638179\">First we rewrite the logarithm in exponential form:[latex]\\,{3}^{y}=\\frac{1}{27}.\\,[\/latex]Next, we ask, \u201cTo what exponent must 3 be raised in order to get[latex]\\,\\frac{1}{27}?[\/latex]\u201d<\/p>\n<p id=\"fs-id1165137552085\">We know[latex]\\,{3}^{3}=27,[\/latex]but what must we do to get the reciprocal,[latex]\\,\\frac{1}{27}?\\,[\/latex]Recall from working with exponents that[latex]\\,{b}^{-a}=\\frac{1}{{b}^{a}}.\\,[\/latex]We use this information to write<\/p>\n\n<div id=\"eip-id1165137550550\" class=\"unnumbered\">[latex]\\begin{array}{l}{3}^{-3}=\\frac{1}{{3}^{3}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{1}{27}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137585807\">Therefore,[latex]\\,{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575754\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_04\">\n<div id=\"fs-id1165137768727\">\n<p id=\"fs-id1165135437134\">Evaluate[latex]\\,y={\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137812401\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137812401\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137812401\"]\n<p id=\"fs-id1165137806792\">[latex]{\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)=-5[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137547253\" class=\"bc-section section\">\n<h3>Using Common Logarithms<\/h3>\n<p id=\"fs-id1165137574205\">Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression[latex]\\,\\mathrm{log}\\left(x\\right)\\,[\/latex]means[latex]\\,{\\mathrm{log}}_{10}\\left(x\\right).\\,[\/latex]We call a base-10 logarithm a <strong>common logarithm<\/strong>. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.<\/p>\n\n<div id=\"fs-id1165137401037\" class=\"textbox key-takeaways\">\n<h3>Definition of the Common Logarithm<\/h3>\n<p id=\"fs-id1165135609332\">A common logarithm is a logarithm with base[latex]\\,10.\\,[\/latex]We write[latex]\\,{\\mathrm{log}}_{10}\\left(x\\right)\\,[\/latex]simply as[latex]\\,\\mathrm{log}\\left(x\\right).\\,[\/latex]The common logarithm of a positive number[latex]\\,x\\,[\/latex]satisfies the following definition.<\/p>\n<p id=\"fs-id1165137601579\">For[latex]\\,x&gt;0,[\/latex]<\/p>\n\n<div id=\"fs-id1165137475905\">[latex]y=\\mathrm{log}\\left(x\\right)\\text{ is equivalent to }{10}^{y}=x[\/latex]<\/div>\n<p id=\"fs-id1165137559681\">We read[latex]\\,\\mathrm{log}\\left(x\\right)\\,[\/latex]as, \u201cthe logarithm with base[latex]\\,10\\,[\/latex]of[latex]\\,x\\,[\/latex]\u201d or \u201clog base 10 of[latex]\\,x.[\/latex]\u201d<\/p>\n<p id=\"fs-id1165137771789\">The logarithm[latex]\\,y\\,[\/latex]is the exponent to which[latex]\\,10\\,[\/latex]must be raised to get[latex]\\,x.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137579434\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137810781\"><strong>Given a common logarithm of the form[latex]\\,y=\\mathrm{log}\\left(x\\right),[\/latex] evaluate it mentally.<\/strong><\/p>\n\n<ol id=\"fs-id1165137828334\" type=\"1\">\n \t<li>Rewrite the argument[latex]\\,x\\,[\/latex]as a power of[latex]\\,10:\\,[\/latex][latex]{10}^{y}=x.[\/latex]<\/li>\n \t<li>Use previous knowledge of powers of[latex]\\,10\\,[\/latex]to identify[latex]\\,y\\,[\/latex]by asking, \u201cTo what exponent must[latex]\\,10\\,[\/latex]be raised in order to get[latex]\\,x?[\/latex]\u201d<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137742366\">\n<div id=\"fs-id1165137418239\">\n<h3>Finding the Value of a Common Logarithm Mentally<\/h3>\n<p id=\"fs-id1165137658546\">Evaluate[latex]\\,y=\\mathrm{log}\\left(1000\\right)\\,[\/latex]without using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137634154\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137634154\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137634154\"]\n<p id=\"fs-id1165137444192\">First we rewrite the logarithm in exponential form:[latex]\\,{10}^{y}=1000.\\,[\/latex]Next, we ask, \u201cTo what exponent must[latex]\\,10\\,[\/latex]be raised in order to get 1000?\u201d We know<\/p>\n\n<div id=\"eip-id1165134331119\" class=\"unnumbered\">[latex]{10}^{3}=1000[\/latex]<\/div>\n<p id=\"fs-id1165137584125\">Therefore,[latex]\\,\\mathrm{log}\\left(1000\\right)=3.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503827\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_05\">\n<div id=\"fs-id1165137673696\">\n<p id=\"fs-id1165137393877\">Evaluate[latex]\\,y=\\mathrm{log}\\left(1,000,000\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137768485\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137768485\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137768485\"]\n<p id=\"fs-id1165137436094\">[latex]\\mathrm{log}\\left(1,000,000\\right)=6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137552804\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137827812\"><strong>Given a common logarithm with the form[latex]\\,y=\\mathrm{log}\\left(x\\right),[\/latex]evaluate it using a calculator.<\/strong><\/p>\n\n<ol id=\"fs-id1165137418685\" type=\"1\">\n \t<li>Press <strong>[LOG]<\/strong>.<\/li>\n \t<li>Enter the value given for[latex]\\,x,[\/latex]followed by <strong>[ ) ]<\/strong>.<\/li>\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137793928\">\n<div id=\"fs-id1165137892249\">\n<h3>Finding the Value of a Common Logarithm Using a Calculator<\/h3>\n<p id=\"fs-id1165137667877\">Evaluate[latex]\\,y=\\mathrm{log}\\left(321\\right)\\,[\/latex]to four decimal places using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137404714\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137404714\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137404714\"]\n<ul id=\"fs-id1165137786486\">\n \t<li>Press <strong>[LOG]<\/strong>.<\/li>\n \t<li>Enter 321<em>,<\/em> followed by <strong>[ ) ]<\/strong>.<\/li>\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137735413\">Rounding to four decimal places,[latex]\\,\\mathrm{log}\\left(321\\right)\\approx 2.5065.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div>\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137789015\">Note that[latex]\\,{10}^{2}=100\\,[\/latex]and that[latex]\\,{10}^{3}=1000.\\,[\/latex]Since 321 is between 100 and 1000, we know that[latex]\\,\\mathrm{log}\\left(321\\right)\\,[\/latex]must be between[latex]\\,\\mathrm{log}\\left(100\\right)\\,[\/latex]and[latex]\\,\\mathrm{log}\\left(1000\\right).\\,[\/latex]This gives us the following:<\/p>\n\n<div id=\"eip-id1165134280435\" class=\"unnumbered\">[latex]\\begin{array}{ccccc}100&amp; &lt;&amp; 321&amp; &lt;&amp; 1000\\\\ 2&amp; &lt;&amp; 2.5065&amp; &lt;&amp; 3\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137780842\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_06\">\n<div id=\"fs-id1165135241210\">\n<p id=\"fs-id1165137735373\">Evaluate[latex]\\,y=\\mathrm{log}\\left(123\\right)\\,[\/latex]to four decimal places using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137550190\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137550190\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137550190\"]\n<p id=\"fs-id1165137844052\">[latex]\\mathrm{log}\\left(123\\right)\\approx 2.0899[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_04_03_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137603561\">\n<div id=\"fs-id1165135704023\">\n<h3>Rewriting and Solving a Real-World Exponential Model<\/h3>\n<p id=\"fs-id1165135194300\">The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation[latex]\\,{10}^{x}=500\\,[\/latex]represents this situation, where[latex]\\,x\\,[\/latex]is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137784516\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137784516\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137784516\"]\n<p id=\"fs-id1165137827621\">We begin by rewriting the exponential equation in logarithmic form.<\/p>\n\n<div id=\"eip-id1165134048114\" class=\"unnumbered\">[latex]\\begin{array}{lll}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{10}^{x}\\hfill &amp; =500\\hfill &amp; \\hfill \\\\ \\mathrm{log}\\left(500\\right)\\hfill &amp; =x\\hfill &amp; \\text{Use the definition of the common log}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137419444\">Next we evaluate the logarithm using a calculator:<\/p>\n\n<ul id=\"fs-id1165137736356\">\n \t<li>Press <strong>[LOG]<\/strong>.<\/li>\n \t<li>Enter[latex]\\,500,[\/latex]followed by <strong>[ ) ]<\/strong>.<\/li>\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\n \t<li>To the nearest thousandth,[latex]\\,\\mathrm{log}\\left(500\\right)\\approx 2.699.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137422793\">The difference in magnitudes was about[latex]\\,2.699.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137749635\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_07\">\n<div id=\"fs-id1165135195254\">\n<p id=\"fs-id1165137736970\">The amount of energy released from one earthquake was[latex]\\,\\text{8,500}\\,[\/latex]times greater than the amount of energy released from another. The equation[latex]\\,{10}^{x}=8500\\,[\/latex]represents this situation, where[latex]\\,x\\,[\/latex]is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137656499\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137656499\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137656499\"]\n<p id=\"fs-id1165137438675\">The difference in magnitudes was about [latex]\\,3.929.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137405741\" class=\"bc-section section\">\n<h3>Using Natural Logarithms<\/h3>\n<p id=\"fs-id1165137661970\">The most frequently used base for logarithms is[latex]\\,e.\\,[\/latex]Base[latex]\\,e\\,[\/latex]logarithms are important in calculus and some scientific applications; they are called <strong>natural logarithms<\/strong>. The base[latex]\\,e\\,[\/latex]logarithm,[latex]\\,{\\mathrm{log}}_{e}\\left(x\\right),[\/latex] has its own notation,[latex]\\,\\mathrm{ln}\\left(x\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137473872\">Most values of[latex]\\,\\mathrm{ln}\\left(x\\right)\\,[\/latex]can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,[latex]\\,\\mathrm{ln}1=0.\\,[\/latex]For other natural logarithms, we can use the[latex]\\,\\mathrm{ln}\\,[\/latex]key that can be found on most scientific calculators. We can also find the natural logarithm of any power of[latex]\\,e\\,[\/latex]using the inverse property of logarithms.<\/p>\n\n<div id=\"fs-id1165137452317\" class=\"textbox key-takeaways\">\n<h3>Definition of the Natural Logarithm<\/h3>\n<p id=\"fs-id1165137579241\">A natural logarithm is a logarithm with base[latex]\\,e.[\/latex] We write [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] simply as [latex]\\mathrm{ln}\\left(x\\right).[\/latex] The natural logarithm of a positive number [latex]x[\/latex] satisfies the following definition.<\/p>\n<p id=\"fs-id1165135613642\">For[latex]\\,x&gt;0,[\/latex]<\/p>\n\n<div id=\"fs-id1165137580230\">[latex]y=\\mathrm{ln}\\left(x\\right)\\text{ is equivalent to }{e}^{y}=x[\/latex]<\/div>\n<p id=\"fs-id1165137658264\">We read[latex]\\,\\mathrm{ln}\\left(x\\right)\\,[\/latex]as, \u201cthe logarithm with base[latex]\\,e\\,[\/latex]of[latex]\\,x[\/latex]\u201d or \u201cthe natural logarithm of[latex]\\,x.[\/latex]\u201d<\/p>\n<p id=\"fs-id1165137566720\">The logarithm[latex]\\,y\\,[\/latex]is the exponent to which[latex]\\,e\\,[\/latex]must be raised to get[latex]\\,x.[\/latex]<\/p>\n<p id=\"fs-id1165137705251\">Since the functions[latex]\\,y=e{}^{x}\\,[\/latex]and[latex]\\,y=\\mathrm{ln}\\left(x\\right)\\,[\/latex]are inverse functions,[latex]\\,\\mathrm{ln}\\left({e}^{x}\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]and[latex]\\,e{}^{\\mathrm{ln}\\left(x\\right)}=x\\,[\/latex]for[latex]\\,x&gt;0.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137409558\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137832169\"><strong>Given a natural logarithm with the form[latex]\\,y=\\mathrm{ln}\\left(x\\right),[\/latex] evaluate it using a calculator.<\/strong><\/p>\n\n<ol id=\"fs-id1165135407195\" type=\"1\">\n \t<li>Press <strong>[LN]<\/strong>.<\/li>\n \t<li>Enter the value given for[latex]\\,x,[\/latex] followed by <strong>[ ) ]<\/strong>.<\/li>\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_03_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137731536\">\n<div id=\"fs-id1165137434974\">\n<h3>Evaluating a Natural Logarithm Using a Calculator<\/h3>\n<p id=\"fs-id1165137573341\">Evaluate[latex]\\,y=\\mathrm{ln}\\left(500\\right)\\,[\/latex]to four decimal places using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137702133\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137702133\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137702133\"]\n<ul id=\"fs-id1165137563770\">\n \t<li>Press <strong>[LN]<\/strong>.<\/li>\n \t<li>Enter[latex]\\,500,[\/latex]followed by <strong>[ ) ]<\/strong>.<\/li>\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137645024\">Rounding to four decimal places,[latex]\\,\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137676028\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_08\">\n<div id=\"fs-id1165137431140\">\n<p id=\"fs-id1165137435623\">Evaluate[latex]\\,\\mathrm{ln}\\left(-500\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137737001\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137737001\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137737001\"]\n<p id=\"fs-id1165137639598\">It is not possible to take the logarithm of a negative number in the set of real numbers.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137648012\" class=\"precalculus media\">\n<p id=\"fs-id1165137451079\">Access this online resource for additional instruction and practice with logarithms.<\/p>\n\n<ul id=\"fs-id1165137732886\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/intrologarithms\">Introduction to Logarithms<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137870892\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id1983134\" summary=\"...\">\n<tbody>\n<tr>\n<td>Definition of the logarithmic function<\/td>\n<td>For[latex]\\text{ } x&gt;0,b&gt;0,b\\ne 1,[\/latex]&lt;[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\text{ }[\/latex]if and only if[latex]\\text{ }{b}^{y}=x.[\/latex]&lt;\/td&gt;<\/td>\n<\/tr>\n<tr>\n<td>Definition of the common logarithm<\/td>\n<td>For[latex]\\text{ }x&gt;0,[\/latex][latex]y=\\mathrm{log}\\left(x\\right)\\text{ }[\/latex]if and only if[latex]\\text{ }{10}^{y}=x.[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Definition of the natural logarithm<\/td>\n<td>For[latex]\\text{ }x&gt;0,[\/latex][latex]y=\\mathrm{ln}\\left(x\\right)\\text{ }[\/latex]if and only if[latex]\\text{ }{e}^{y}=x.[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135699130\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137574258\">\n \t<li>The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.<\/li>\n \t<li>Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See <a class=\"autogenerated-content\" href=\"#Example_04_03_01\">(Figure)<\/a>.<\/li>\n \t<li>Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See <a class=\"autogenerated-content\" href=\"#Example_04_03_02\">(Figure)<\/a>.<\/li>\n \t<li>Logarithmic functions with base[latex]\\,b\\,[\/latex]can be evaluated mentally using previous knowledge of powers of[latex]\\,b.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_03_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_04_03_04\">(Figure)<\/a>.<\/li>\n \t<li>Common logarithms can be evaluated mentally using previous knowledge of powers of[latex]\\,10.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_03_05\">(Figure)<\/a><strong>.<\/strong><\/li>\n \t<li>When common logarithms cannot be evaluated mentally, a calculator can be used. See <a class=\"autogenerated-content\" href=\"#Example_04_03_06\">(Figure)<\/a><strong>.<\/strong><\/li>\n \t<li>Real-world exponential problems with base[latex]\\,10\\,[\/latex]can be rewritten as a common logarithm and then evaluated using a calculator. See <a class=\"autogenerated-content\" href=\"#Example_04_03_07\">(Figure)<\/a><strong>.<\/strong><\/li>\n \t<li>Natural logarithms can be evaluated using a calculator <a class=\"autogenerated-content\" href=\"#Example_04_03_08\">(Figure)<\/a><strong>.<\/strong><\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135192789\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137427076\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137817361\">\n<div id=\"fs-id1165137559978\">\n<p id=\"fs-id1165137445179\">What is a base[latex]\\,b\\,[\/latex]logarithm? Discuss the meaning by interpreting each part of the equivalent equations[latex]\\,{b}^{y}=x\\,[\/latex]and[latex]\\,{\\mathrm{log}}_{b}x=y\\,[\/latex]for[latex]\\,b&gt;0,b\\ne 1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137411524\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137411524\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137411524\"]A logarithm is an exponent. Specifically, it is the exponent to which a base[latex]\\,b\\,[\/latex]is raised to produce a given value. In the expressions given, the base[latex]\\,b\\,[\/latex]has the same value. The exponent,[latex]\\,y,[\/latex]in the expression[latex]\\,{b}^{y}\\,[\/latex]can also be written as the logarithm,[latex]\\,{\\mathrm{log}}_{b}x,[\/latex]and the value of[latex]\\,x\\,[\/latex]is the result of raising[latex]\\,b\\,[\/latex]to the power of[latex]\\,y.[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137574896\">\n<div id=\"fs-id1165137658231\">\n<p id=\"fs-id1165137529141\">How is the logarithmic function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}x\\,[\/latex]related to the exponential function[latex]\\,g\\left(x\\right)={b}^{x}?\\,[\/latex]What is the result of composing these two functions?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137446568\">\n<div id=\"fs-id1165137602953\">\n<p id=\"fs-id1165137557324\">How can the logarithmic equation[latex]\\,{\\mathrm{log}}_{b}x=y\\,[\/latex]be solved for[latex]\\,x\\,[\/latex]using the properties of exponents?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135203815\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135203815\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135203815\"]\n<p id=\"fs-id1165137530592\">Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation[latex]\\,{b}^{y}=x,[\/latex] and then properties of exponents can be applied to solve for[latex]\\,x.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137470358\">\n<div id=\"fs-id1165135526986\">\n<p id=\"fs-id1165137697077\">Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base[latex]\\,b,[\/latex] and how does the notation differ?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137507578\">\n<div id=\"fs-id1165137828407\">\n<p id=\"fs-id1165135417853\">Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base[latex]\\,b,[\/latex] and how does the notation differ?<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137596568\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137596568\"]\n<p id=\"fs-id1165137596568\">The natural logarithm is a special case of the logarithm with base[latex]\\,b\\,[\/latex]in that the natural log always has base[latex]\\,e.\\,[\/latex]Rather than notating the natural logarithm as[latex]\\,{\\mathrm{log}}_{e}\\left(x\\right),[\/latex]the notation used is[latex]\\,\\mathrm{ln}\\left(x\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447239\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137414571\">For the following exercises, rewrite each equation in exponential form.<\/p>\n\n<div id=\"fs-id1165137646887\">\n<div id=\"fs-id1165137664870\">\n<p id=\"fs-id1165137726644\">[latex]{\\text{log}}_{4}\\left(q\\right)=m[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137473551\">\n<div id=\"fs-id1165137454663\">\n<p id=\"fs-id1165137736589\">[latex]{\\text{log}}_{a}\\left(b\\right)=c[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135191877\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135191877\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135191877\"]\n<p id=\"fs-id1165137531424\">[latex]{a}^{c}=b[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137506749\">\n<div id=\"fs-id1165135536326\">\n<p id=\"fs-id1165137430583\">[latex]{\\mathrm{log}}_{16}\\left(y\\right)=x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410961\">\n<div id=\"fs-id1165137602136\">\n<p id=\"fs-id1165137654948\">[latex]{\\mathrm{log}}_{x}\\left(64\\right)=y[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137428120\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137428120\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137428120\"]\n<p id=\"fs-id1165135192141\">[latex]{x}^{y}=64[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673422\">\n<div id=\"fs-id1165135585639\">\n<p id=\"fs-id1165135516961\">[latex]{\\mathrm{log}}_{y}\\left(x\\right)=-11[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135356538\">\n<div id=\"fs-id1165137501385\">\n<p id=\"fs-id1165137629520\">[latex]{\\mathrm{log}}_{15}\\left(a\\right)=b[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135191871\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135191871\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135191871\"]\n[latex]{15}^{b}=a[\/latex][\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137580855\">\n<div id=\"fs-id1165137438413\">\n<p id=\"fs-id1165137656672\">[latex]{\\mathrm{log}}_{y}\\left(137\\right)=x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137611652\">\n<div id=\"fs-id1165134038222\">\n<p id=\"fs-id1165137645621\">[latex]{\\mathrm{log}}_{13}\\left(142\\right)=a[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137580134\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137580134\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137580134\"]\n<p id=\"fs-id1165135174998\">[latex]{13}^{a}=142[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137724172\">\n<div id=\"fs-id1165137724174\">\n<p id=\"fs-id1165134077330\">[latex]\\text{log}\\left(v\\right)=t[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735890\">\n<div id=\"fs-id1165137735892\">\n<p id=\"fs-id1165137560511\">[latex]\\text{ln}\\left(w\\right)=n[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135185210\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135185210\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135185210\"]\n<p id=\"fs-id1165135185212\">[latex]{e}^{n}=w[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137810335\">For the following exercises, rewrite each equation in logarithmic form.<\/p>\n\n<div id=\"fs-id1165135186264\">\n<div id=\"fs-id1165135203858\">\n<p id=\"fs-id1165135203860\">[latex]{4}^{x}=y[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135194695\">\n<div id=\"fs-id1165137724816\">\n<p id=\"fs-id1165137724818\">[latex]{c}^{d}=k[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137742265\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137742265\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137742265\"]\n<p id=\"fs-id1165137742268\">[latex]{\\text{log}}_{c}\\left(k\\right)=d[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137698394\">\n<div id=\"fs-id1165137434093\">\n<p id=\"fs-id1165137434095\">[latex]{m}^{-7}=n[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135384408\">\n<div id=\"fs-id1165135384410\">\n<p id=\"fs-id1165137534920\">[latex]{19}^{x}=y[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135296129\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135296129\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135296129\"]\n<p id=\"fs-id1165137771226\">[latex]{\\mathrm{log}}_{19}y=x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135358961\">\n<div id=\"fs-id1165137627327\">\n<p id=\"fs-id1165137911046\">[latex]{x}^{-\\,\\frac{10}{13}}=y[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132971714\">\n<div id=\"fs-id1165132971716\">\n<p id=\"fs-id1165137728409\">[latex]{n}^{4}=103[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134032258\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134032258\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134032258\"]\n<p id=\"fs-id1165137447080\">[latex]{\\mathrm{log}}_{n}\\left(103\\right)=4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137564489\">\n<div id=\"fs-id1165137564491\">\n<p id=\"fs-id1165137404398\">[latex]{\\left(\\frac{7}{5}\\right)}^{m}=n[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731983\">\n<div id=\"fs-id1165137597971\">\n<p id=\"fs-id1165137597973\">[latex]{y}^{x}=\\frac{39}{100}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135194466\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135194466\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135194466\"]\n<p id=\"fs-id1165135194469\">[latex]{\\mathrm{log}}_{y}\\left(\\frac{39}{100}\\right)=x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137827739\">\n<div id=\"fs-id1165137762434\">\n<p id=\"fs-id1165137762436\">[latex]{10}^{a}=b[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135296148\">\n<div id=\"fs-id1165134040600\">\n<p id=\"fs-id1165134040602\">[latex]{e}^{k}=h[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135191801\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135191801\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135191801\"]\n<p id=\"fs-id1165135191803\">[latex]\\text{ln}\\left(h\\right)=k[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137762947\">For the following exercises, solve for[latex]\\,x\\,[\/latex]by converting the logarithmic equation to exponential form.<\/p>\n\n<div>\n<div id=\"fs-id1165137438779\">\n<p id=\"fs-id1165137823075\">[latex]{\\text{log}}_{3}\\left(x\\right)=2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137482919\">\n<div id=\"fs-id1165137628211\">\n<p id=\"fs-id1165137628213\">[latex]{\\text{log}}_{2}\\left(x\\right)=-3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137443266\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137443266\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137443266\"]\n<p id=\"fs-id1165137834752\">[latex]x={2}^{-3}=\\frac{1}{8}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137726345\">\n<div id=\"fs-id1165137726348\">\n<p id=\"fs-id1165137473443\">[latex]{\\text{log}}_{5}\\left(x\\right)=2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137389073\">\n<div id=\"fs-id1165137639044\">\n<p id=\"fs-id1165137639046\">[latex]{\\mathrm{log}}_{3}\\left(x\\right)=3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137409760\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137409760\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137409760\"]\n<p id=\"fs-id1165137409762\">[latex]x={3}^{3}=27[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661950\">\n<div id=\"fs-id1165137726952\">\n<p id=\"fs-id1165137475701\">[latex]{\\text{log}}_{2}\\left(x\\right)=6[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627690\">\n<div>\n<p id=\"fs-id1165137444979\">[latex]{\\text{log}}_{9}\\left(x\\right)=\\frac{1}{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137469004\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137469004\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137469004\"]\n<p id=\"fs-id1165137651832\">[latex]x={9}^{\\frac{1}{2}}=3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203739\">\n<div>\n<p id=\"fs-id1165137427716\">[latex]{\\text{log}}_{18}\\left(x\\right)=2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613168\">\n<div id=\"fs-id1165137414109\">\n<p id=\"fs-id1165137414111\">[latex]{\\mathrm{log}}_{6}\\left(x\\right)=-3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135253754\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135253754\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135253754\"]\n<p id=\"fs-id1165137543450\">[latex]x={6}^{-3}=\\frac{1}{216}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137937666\">\n<div id=\"fs-id1165137937668\">\n<p id=\"fs-id1165137758189\">[latex]\\text{log}\\left(x\\right)=3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736080\">\n<div id=\"fs-id1165137442469\">\n<p id=\"fs-id1165137442471\">[latex]\\text{ln}\\left(x\\right)=2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135524549\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135524549\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135524549\"]\n<p id=\"fs-id1165135524551\">[latex]x={e}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137634430\">For the following exercises, use the definition of common and natural logarithms to simplify.<\/p>\n\n<div id=\"fs-id1165137634434\">\n<div id=\"fs-id1165137732545\">\n<p id=\"fs-id1165137732547\">[latex]\\text{log}\\left({100}^{8}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137936925\">\n<div id=\"fs-id1165137936927\">\n<p id=\"fs-id1165135526105\">[latex]{10}^{\\text{log}\\left(32\\right)}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137436613\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137436613\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137436613\"]\n<p id=\"fs-id1165137942473\">[latex]32[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137529000\">\n<div id=\"fs-id1165137529002\">\n<p id=\"fs-id1165135152291\">[latex]2\\text{log}\\left(.0001\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134079645\">\n<div id=\"fs-id1165137566210\">\n<p id=\"fs-id1165137566212\">[latex]{e}^{\\mathrm{ln}\\left(1.06\\right)}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137455559\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137455559\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137455559\"]\n<p id=\"fs-id1165137464570\">[latex]1.06[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137400162\">\n<div id=\"fs-id1165137400164\">\n<p id=\"fs-id1165135259620\">[latex]\\mathrm{ln}\\left({e}^{-5.03}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137543862\">\n<div id=\"fs-id1165137543865\">\n<p id=\"fs-id1165135434891\">[latex]{e}^{\\mathrm{ln}\\left(10.125\\right)}+4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137605063\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137605063\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137605063\"]\n[latex]14.125[\/latex][\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451809\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137564770\">For the following exercises, evaluate the base[latex]\\,b\\,[\/latex]logarithmic expression without using a calculator.<\/p>\n\n<div id=\"fs-id1165137676437\">\n<div id=\"fs-id1165137676439\">\n<p id=\"fs-id1165137728122\">[latex]{\\text{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137590077\">\n<div id=\"fs-id1165137590079\">\n<p id=\"fs-id1165134108525\">[latex]{\\text{log}}_{6}\\left(\\sqrt{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135706822\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135706822\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135706822\"]\n<p id=\"fs-id1165137911553\">[latex]\\frac{1}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137415151\">\n<div id=\"fs-id1165137423807\">\n<p id=\"fs-id1165137423810\">[latex]{\\text{log}}_{2}\\left(\\frac{1}{8}\\right)+4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190140\">\n<div id=\"fs-id1165137806053\">\n<p id=\"fs-id1165137806055\">[latex]6{\\text{log}}_{8}\\left(4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137454533\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137454533\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137454533\"]\n<p id=\"fs-id1165137454536\">[latex]4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137532658\">For the following exercises, evaluate the common logarithmic expression without using a calculator.<\/p>\n\n<div id=\"fs-id1165137542512\">\n<div id=\"fs-id1165135613435\">\n<p id=\"fs-id1165135613437\">[latex]\\text{log}\\left(10,000\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137755960\">\n<div id=\"fs-id1165137736340\">\n<p id=\"fs-id1165137736342\">[latex]\\text{log}\\left(0.001\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135431079\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135431079\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135431079\"]\n<p id=\"fs-id1165135431081\">[latex]-\\text{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190197\">\n<div id=\"fs-id1165135190199\">\n<p id=\"fs-id1165137422196\">[latex]\\text{log}\\left(1\\right)+7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137432736\">\n<div id=\"fs-id1165137655695\">\n<p id=\"fs-id1165137655697\">[latex]2\\text{log}\\left({100}^{-3}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137588216\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137588216\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137588216\"]\n<p id=\"fs-id1165137588218\">[latex]-12[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137570442\">For the following exercises, evaluate the natural logarithmic expression without using a calculator.<\/p>\n\n<div id=\"fs-id1165137619029\">\n<div id=\"fs-id1165137619032\">\n<p id=\"fs-id1165134047613\">[latex]\\text{ln}\\left({e}^{\\frac{1}{3}}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736480\">\n<div id=\"fs-id1165137736482\">\n<p id=\"fs-id1165137645639\">[latex]\\text{ln}\\left(1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137645466\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137645466\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137645466\"]\n<p id=\"fs-id1165135207524\">[latex]0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137804477\">\n<div id=\"fs-id1165137474568\">\n<p id=\"fs-id1165137474570\">[latex]\\text{ln}\\left({e}^{-0.225}\\right)-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137762957\">\n<div id=\"fs-id1165135570229\">\n<p id=\"fs-id1165135570231\">[latex]25\\text{ln}\\left({e}^{\\frac{2}{5}}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137418633\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137418633\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137418633\"]\n<p id=\"fs-id1165137418635\">[latex]10[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137527430\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137401108\">For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.<\/p>\n\n<div id=\"fs-id1165137626617\">\n<div id=\"fs-id1165137626619\">\n<p id=\"fs-id1165137793717\">[latex]\\text{log}\\left(0.04\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640432\">\n<div id=\"fs-id1165137640434\">\n<p id=\"fs-id1165135194090\">[latex]\\text{ln}\\left(15\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137696603\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137696603\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137696603\"]\n<p id=\"fs-id1165137679201\">[latex]\\text{2}.\\text{7}0\\text{8}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731720\">\n<div id=\"fs-id1165137731722\">\n<p id=\"fs-id1165137463176\">[latex]\\text{ln}\\left(\\frac{4}{5}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137527828\">\n<div id=\"fs-id1165137527830\">\n<p id=\"fs-id1165137657921\">[latex]\\text{log}\\left(\\sqrt{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137658294\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137658294\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137658294\"]\n<p id=\"fs-id1165137405172\">[latex]0.151[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137655087\">\n<div id=\"fs-id1165137655089\">\n<p id=\"fs-id1165137757908\">[latex]\\text{ln}\\left(\\sqrt{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932472\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137435099\">\n<div id=\"fs-id1165137593336\">\n<p id=\"fs-id1165137593338\">Is[latex]\\,x=0\\,[\/latex]in the domain of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(x\\right)?\\,[\/latex]If so, what is the value of the function when[latex]\\,x=0?\\,[\/latex]Verify the result.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137474123\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137474123\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137474123\"]\n<p id=\"fs-id1165137911451\">No, the function has no defined value for[latex]\\,x=0.\\,[\/latex]To verify, suppose[latex]\\,x=0\\,[\/latex]is in the domain of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(x\\right).\\,[\/latex]Then there is some number[latex]\\,n\\,[\/latex]such that[latex]\\,n=\\mathrm{log}\\left(0\\right).\\,[\/latex]Rewriting as an exponential equation gives:[latex]\\,{10}^{n}=0,[\/latex] which is impossible since no such real number[latex]\\,n\\,[\/latex]exists. Therefore,[latex]\\,x=0\\,[\/latex]is <em>not<\/em> the domain of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(x\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137597455\">\n<div id=\"fs-id1165137597458\">\n<p id=\"fs-id1165137627827\">Is[latex]\\,f\\left(x\\right)=0\\,[\/latex]in the range of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(x\\right)?\\,[\/latex]If so, for what value of[latex]\\,x?\\,[\/latex]Verify the result.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137548869\">\n<div id=\"fs-id1165137548871\">\n<p id=\"fs-id1165135613701\">Is there a number[latex]\\,x\\,[\/latex]such that[latex]\\,\\mathrm{ln}x=2?\\,[\/latex]If so, what is that number? Verify the result.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137659786\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137659786\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137659786\"]\n<p id=\"fs-id1165137806408\">Yes. Suppose there exists a real number[latex]\\,x\\,[\/latex]such that[latex]\\,\\mathrm{ln}x=2.\\,[\/latex]Rewriting as an exponential equation gives[latex]\\,x={e}^{2},[\/latex] which is a real number. To verify, let[latex]\\,x={e}^{2}.\\,[\/latex]Then, by definition,[latex]\\,\\mathrm{ln}\\left(x\\right)=\\mathrm{ln}\\left({e}^{2}\\right)=2.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640809\">\n<div id=\"fs-id1165137603717\">\n<p id=\"fs-id1165137603719\">Is the following true:[latex]\\,\\frac{{\\mathrm{log}}_{3}\\left(27\\right)}{{\\mathrm{log}}_{4}\\left(\\frac{1}{64}\\right)}=-1?\\,[\/latex]Verify the result.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137453751\">\n<div id=\"fs-id1165137453754\">\n<p id=\"fs-id1165137413735\">Is the following true:[latex]\\,\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}=1.725?\\,[\/latex]Verify the result.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137749907\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137749907\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137749907\"]\n<p id=\"fs-id1165137559774\">No;[latex]\\,\\mathrm{ln}\\left(1\\right)=0,[\/latex] so[latex]\\,\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}\\,[\/latex]is undefined.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135437184\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165137732791\">\n<div id=\"fs-id1165137913993\">\n<p id=\"fs-id1165137913995\">The exposure index[latex]\\,EI\\,[\/latex]for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation[latex]\\,EI={\\mathrm{log}}_{2}\\left(\\frac{{f}^{2}}{t}\\right),[\/latex] where[latex]\\,f\\,[\/latex]is the \u201cf-stop\u201d setting on the camera, and [latex]t[\/latex] is the exposure time in seconds. Suppose the f-stop setting is[latex]\\,8\\,[\/latex]and the desired exposure time is[latex]\\,2\\,[\/latex]seconds. What will the resulting exposure index be?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137602359\">\n<div id=\"fs-id1165137838750\">\n<p id=\"fs-id1165137838752\">Refer to the previous exercise. Suppose the light meter on a camera indicates an[latex]\\,EI\\,[\/latex]of[latex]\\,-2,[\/latex] and the desired exposure time is 16 seconds. What should the f-stop setting be?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137570610\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137570610\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137570610\"]\n<p id=\"fs-id1165137417561\">[latex]2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137697043\">The intensity levels <em>I<\/em> of two earthquakes measured on a seismograph can be compared by the formula[latex]\\,\\mathrm{log}\\frac{{I}_{1}}{{I}_{2}}={M}_{1}-{M}_{2}\\,[\/latex]where[latex]\\,M\\,[\/latex]is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.[footnote]<a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/world\/historical.php\">http:\/\/earthquake.usgs.gov\/earthquakes\/world\/historical.php<\/a>. Accessed 3\/4\/2014.[\/footnote] How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135160066\">\n \t<dt>common logarithm<\/dt>\n \t<dd id=\"fs-id1165137571387\">the exponent to which 10 must be raised to get[latex]\\,x;\\,[\/latex][latex]\\,{\\mathrm{log}}_{10}\\left(x\\right)\\,[\/latex] is written simply as[latex]\\,\\mathrm{log}\\left(x\\right).[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137780762\">\n \t<dt>logarithm<\/dt>\n \t<dd id=\"fs-id1165137849198\">the exponent to which[latex]\\,b\\,[\/latex]must be raised to get[latex]\\,x;\\,[\/latex]written[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl>\n \t<dt>natural logarithm<\/dt>\n \t<dd>the exponent to which the number[latex]\\,e\\,[\/latex]must be raised to get[latex]\\,x;[\/latex][latex]\\,{\\mathrm{log}}_{e}\\left(x\\right)\\,[\/latex]is written as[latex]\\,\\mathrm{ln}\\left(x\\right).[\/latex]<\/dd>\n<\/dl>\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>Convert from logarithmic to exponential form.<\/li>\n<li>Convert from exponential to logarithmic form.<\/li>\n<li>Evaluate logarithms.<\/li>\n<li>Use common logarithms.<\/li>\n<li>Use natural logarithms.<\/li>\n<\/ul>\n<\/div>\n<div id=\"CNX_Precalc_Figure_04_03_001\" class=\"small aligncenter\">\n<figure style=\"width: 488px\" 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\/19140521\/CNX_Precalc_Figure_04_03_001.jpg\" alt=\"Photo of the aftermath of the earthquake in Japan with a focus on the Japanese flag.\" width=\"488\" height=\"325\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)<\/figcaption><\/figure>\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\n<p id=\"fs-id1165137557013\">In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary. Accessed 3\/4\/2013.\" id=\"return-footnote-102-1\" href=\"#footnote-102-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> . One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary. Accessed 3\/4\/2013.\" id=\"return-footnote-102-2\" href=\"#footnote-102-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a> like those shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_03_001\">(Figure)<\/a>. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/. Accessed 3\/4\/2013.\" id=\"return-footnote-102-3\" href=\"#footnote-102-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a> whereas the Japanese earthquake registered a 9.0.<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/. Accessed 3\/4\/2013.\" id=\"return-footnote-102-4\" href=\"#footnote-102-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a><\/p>\n<p id=\"fs-id1165137760714\">The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is [latex]{10}^{8-4}={10}^{4}=10,000[\/latex] times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.<\/p>\n<div id=\"fs-id1165137644550\" class=\"bc-section section\">\n<h3>Converting from Logarithmic to Exponential Form<\/h3>\n<p id=\"fs-id1165135192781\">In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is[latex]\\,{10}^{x}=500,[\/latex] where[latex]\\,x\\,[\/latex]represents the difference in magnitudes on the <span class=\"no-emphasis\">Richter Scale<\/span>. How would we solve for[latex]\\,x?[\/latex]<\/p>\n<p id=\"fs-id1165135160312\">We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve[latex]\\,{10}^{x}=500.\\,[\/latex]We know that[latex]\\,{10}^{2}=100\\,[\/latex]and[latex]\\,{10}^{3}=1000,[\/latex] so it is clear that[latex]\\,x\\,[\/latex]must be some value between 2 and 3, since[latex]\\,y={10}^{x}\\,[\/latex]is increasing. We can examine a graph, as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_03_002\">(Figure)<\/a><strong>,<\/strong> to better estimate the solution.<\/p>\n<div id=\"CNX_Precalc_Figure_04_03_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\/19140524\/CNX_Precalc_Figure_04_03_002.jpg\" alt=\"Graph of the intersections of the equations y=10^x and y=500.\" width=\"487\" height=\"477\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137662989\">Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_03_002\">(Figure)<\/a> passes the horizontal line test. The exponential function[latex]\\,y={b}^{x}\\,[\/latex]is <span class=\"no-emphasis\">one-to-one<\/span>, so its inverse,[latex]\\,x={b}^{y}\\,[\/latex]is also a function. As is the case with all inverse functions, we simply interchange[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]and solve for[latex]\\,y\\,[\/latex]to find the inverse function. To represent[latex]\\,y\\,[\/latex]as a function of[latex]\\,x,[\/latex] we use a logarithmic function of the form[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right).\\,[\/latex]The base[latex]\\,b\\,[\/latex]<strong>logarithm<\/strong> of a number is the exponent by which we must raise[latex]\\,b\\,[\/latex]to get that number.<\/p>\n<p id=\"fs-id1165137404844\">We read a logarithmic expression as, \u201cThe logarithm with base[latex]\\,b\\,[\/latex]of[latex]\\,x\\,[\/latex]is equal to[latex]\\,y,[\/latex]\u201d or, simplified, \u201clog base[latex]\\,b\\,[\/latex]of[latex]\\,x\\,[\/latex]is[latex]\\,y.[\/latex]\u201d We can also say, \u201c[latex]b\\,[\/latex]raised to the power of[latex]\\,y\\,[\/latex]is[latex]\\,x,[\/latex]\u201d because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since[latex]\\,{2}^{5}=32,[\/latex] we can write[latex]\\,{\\mathrm{log}}_{2}32=5.\\,[\/latex]We read this as \u201clog base 2 of 32 is 5.\u201d<\/p>\n<p id=\"fs-id1165137597501\">We can express the relationship between logarithmic form and its corresponding exponential form as follows:<\/p>\n<div id=\"eip-604\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y\u21d4{b}^{y}=x,\\text{}b>0,b\\ne 1[\/latex]<\/div>\n<p id=\"fs-id1165137678993\">Note that the base[latex]\\,b\\,[\/latex]is always positive.<\/p>\n<p><span id=\"fs-id1165137696233\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140526\/CNX_Precalc_Figure_04_03_004.jpg\" alt=\"\" \/><\/span><\/p>\n<p>Because logarithm is a function, it is most correctly written as[latex]\\,{\\mathrm{log}}_{b}\\left(x\\right),[\/latex] using parentheses to denote function evaluation, just as we would with[latex]\\,f\\left(x\\right).\\,[\/latex]However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as[latex]\\,{\\mathrm{log}}_{b}x.\\,[\/latex]Note that many calculators require parentheses around the[latex]\\,x.[\/latex]<\/p>\n<p id=\"fs-id1165137827516\">We can illustrate the notation of logarithms as follows:<\/p>\n<p><span id=\"fs-id1165137771679\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140532\/CNX_Precalc_Figure_04_03_003.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137575165\">Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]and[latex]\\,y={b}^{x}\\,[\/latex]are inverse functions.<\/p>\n<div id=\"fs-id1165137472937\" class=\"textbox key-takeaways\">\n<h3>Definition of the Logarithmic Function<\/h3>\n<p id=\"fs-id1165137704597\">A logarithm base[latex]\\,b\\,[\/latex]of a positive number[latex]\\,x\\,[\/latex]satisfies the following definition.<\/p>\n<p id=\"fs-id1165137584967\">For[latex]\\,x>0,b>0,b\\ne 1,[\/latex]<\/p>\n<div id=\"fs-id1165137433829\">[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\text{ is equivalent to }{b}^{y}=x[\/latex]<\/div>\n<p id=\"fs-id1165137893373\">where,<\/p>\n<ul id=\"fs-id1165135530561\">\n<li>we read[latex]\\,{\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]as, \u201cthe logarithm with base[latex]\\,b\\,[\/latex]of[latex]\\,x[\/latex]\u201d or the \u201clog base[latex]\\,b\\,[\/latex]of[latex]\\,x.\"[\/latex]<\/li>\n<li>the logarithm[latex]\\,y\\,[\/latex]is the exponent to which[latex]\\,b\\,[\/latex]must be raised to get[latex]\\,x.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137547773\">Also, since the logarithmic and exponential functions switch the[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,<\/p>\n<ul id=\"fs-id1165137643167\">\n<li>the domain of the logarithm function with base[latex]\\,b \\text{is} \\left(0,\\infty \\right).[\/latex]<\/li>\n<li>the range of the logarithm function with base[latex]\\,b \\text{is} \\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137677696\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1549475\"><strong>Can we take the logarithm of a negative number?<\/strong><\/p>\n<p id=\"fs-id1165137653864\"><em>No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165137874700\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137806301\"><strong>Given an equation in logarithmic form<\/strong>[latex]\\,{\\mathrm{log}}_{b}\\left(x\\right)=y,[\/latex]<strong> convert it to exponential form.<\/strong><\/p>\n<ol id=\"fs-id1165137641669\" type=\"1\">\n<li>Examine the equation[latex]\\,y={\\mathrm{log}}_{b}x\\,[\/latex]and identify[latex]\\,b,y,\\text{and}x.[\/latex]<\/li>\n<li>Rewrite[latex]\\,{\\mathrm{log}}_{b}x=y\\,[\/latex]as[latex]\\,{b}^{y}=x.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135570363\">\n<div id=\"fs-id1165137557855\">\n<h3>Converting from Logarithmic Form to Exponential Form<\/h3>\n<p id=\"fs-id1165137580570\">Write the following logarithmic equations in exponential form.<\/p>\n<ol id=\"fs-id1165137705346\" type=\"a\">\n<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]<\/li>\n<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135613330\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137408172\">First, identify the values of[latex]\\,b,y,\\text{and}x.\\,[\/latex]Then, write the equation in the form[latex]\\,{b}^{y}=x.[\/latex]<\/p>\n<ol id=\"fs-id1165137705659\" type=\"a\">\n<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]\n<p id=\"fs-id1165137602796\">Here,[latex]\\,b=6,y=\\frac{1}{2},\\text{and} x=\\sqrt{6.}\\,[\/latex]Therefore, the equation[latex]\\,{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}\\,[\/latex]is equivalent to[latex]\\,{6}^{\\frac{1}{2}}=\\sqrt{6}.[\/latex]<\/p>\n<\/li>\n<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]\n<p id=\"fs-id1165137698078\">Here,[latex]\\,b=3,y=2,\\text{and} x=9.\\,[\/latex]Therefore, the equation[latex]\\,{\\mathrm{log}}_{3}\\left(9\\right)=2\\,[\/latex]is equivalent to[latex]\\,{3}^{2}=9.\\,[\/latex]<\/p>\n<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640140\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165135208926\">\n<p id=\"fs-id1165137418681\">Write the following logarithmic equations in exponential form.<\/p>\n<ol id=\"fs-id1165137772342\" type=\"a\">\n<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex]<\/li>\n<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135195688\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165137414337\" type=\"a\">\n<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6\\,[\/latex]is equivalent to[latex]\\,{10}^{6}=1,000,000[\/latex]<\/li>\n<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2\\,[\/latex]is equivalent to[latex]\\,{5}^{2}=25[\/latex]<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137585244\" class=\"bc-section section\">\n<h3>Converting from Exponential to Logarithmic Form<\/h3>\n<p id=\"fs-id1165137933968\">To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base[latex]\\,b,[\/latex]exponent[latex]\\,x,[\/latex]and output[latex]\\,y.\\,[\/latex]Then we write[latex]\\,x={\\mathrm{log}}_{b}\\left(y\\right).[\/latex]<\/p>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135168111\">\n<div id=\"fs-id1165137727912\">\n<h3>Converting from Exponential Form to Logarithmic Form<\/h3>\n<p id=\"fs-id1165137804412\">Write the following exponential equations in logarithmic form.<\/p>\n<ol id=\"fs-id1165135192287\" type=\"a\">\n<li>[latex]{2}^{3}=8[\/latex]<\/li>\n<li>[latex]{5}^{2}=25[\/latex]<\/li>\n<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137702205\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137474116\">First, identify the values of[latex]\\,b,y,\\text{and}x.\\,[\/latex]Then, write the equation in the form[latex]\\,x={\\mathrm{log}}_{b}\\left(y\\right).[\/latex]<\/p>\n<ol id=\"fs-id1165137573458\" type=\"a\">\n<li>[latex]{2}^{3}=8[\/latex]\n<p id=\"fs-id1165137466396\">Here,[latex]\\,b=2,[\/latex][latex]\\,x=3,[\/latex]and[latex]\\,y=8.\\,[\/latex]Therefore, the equation[latex]\\,{2}^{3}=8\\,[\/latex]is equivalent to[latex]\\,{\\mathrm{log}}_{2}\\left(8\\right)=3.[\/latex]<\/p>\n<\/li>\n<li>[latex]{5}^{2}=25[\/latex]\n<p id=\"fs-id1165135193035\">Here,[latex]\\,b=5,[\/latex][latex]\\,x=2,[\/latex]and[latex]\\,y=25.\\,[\/latex]Therefore, the equation[latex]\\,{5}^{2}=25\\,[\/latex]is equivalent to[latex]\\,{\\mathrm{log}}_{5}\\left(25\\right)=2.[\/latex]<\/p>\n<\/li>\n<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]\n<p id=\"fs-id1165135187822\">Here,[latex]\\,b=10,[\/latex][latex]\\,x=-4,[\/latex]and[latex]\\,y=\\frac{1}{10,000}.\\,[\/latex]Therefore, the equation[latex]\\,{10}^{-4}=\\frac{1}{10,000}\\,[\/latex]is equivalent to[latex]\\,{\\text{log}}_{10}\\left(\\frac{1}{10,000}\\right)=-4.[\/latex]<\/p>\n<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438165\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165135190969\">\n<p id=\"fs-id1165137566762\">Write the following exponential equations in logarithmic form.<\/p>\n<ol id=\"fs-id1165137771963\" type=\"a\">\n<li>[latex]{3}^{2}=9[\/latex]<\/li>\n<li>[latex]{5}^{3}=125[\/latex]<\/li>\n<li>[latex]{2}^{-1}=\\frac{1}{2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165134065138\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165137469846\" type=\"a\">\n<li>[latex]{3}^{2}=9\\,[\/latex]is equivalent to[latex]\\,{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]<\/li>\n<li>[latex]{5}^{3}=125\\,[\/latex]is equivalent to[latex]\\,{\\mathrm{log}}_{5}\\left(125\\right)=3[\/latex]<\/li>\n<li>[latex]{2}^{-1}=\\frac{1}{2}\\,[\/latex]is equivalent to[latex]\\,{\\text{log}}_{2}\\left(\\frac{1}{2}\\right)=-1[\/latex]<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137530906\" class=\"bc-section section\">\n<h3>Evaluating Logarithms<\/h3>\n<p id=\"fs-id1165137422589\">Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider[latex]\\,{\\mathrm{log}}_{2}8.\\,[\/latex]We ask, \u201cTo what exponent must [latex]\\,2\\,[\/latex] be raised in order to get 8?\u201d Because we already know[latex]\\,{2}^{3}=8,[\/latex] it follows that[latex]\\,{\\mathrm{log}}_{2}8=3.[\/latex]<\/p>\n<p id=\"fs-id1165137733822\">Now consider solving[latex]\\,{\\mathrm{log}}_{7}49\\,[\/latex]and[latex]\\,{\\mathrm{log}}_{3}27\\,[\/latex]mentally.<\/p>\n<ul id=\"fs-id1165137937690\">\n<li>We ask, \u201cTo what exponent must 7 be raised in order to get 49?\u201d We know[latex]\\,{7}^{2}=49.\\,[\/latex]Therefore,[latex]\\,{\\mathrm{log}}_{7}49=2[\/latex]<\/li>\n<li>We ask, \u201cTo what exponent must 3 be raised in order to get 27?\u201d We know[latex]\\,{3}^{3}=27.\\,[\/latex]Therefore,[latex]\\,{\\mathrm{log}}_{3}27=3[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137456358\">Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate[latex]\\,{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}\\,[\/latex]mentally.<\/p>\n<ul id=\"fs-id1165137584208\">\n<li>We ask, \u201cTo what exponent must[latex]\\,\\frac{2}{3}\\,[\/latex]be raised in order to get[latex]\\,\\frac{4}{9}?\\,[\/latex]\u201d We know[latex]\\,{2}^{2}=4\\,[\/latex]and[latex]\\,{3}^{2}=9,[\/latex]so[latex]\\,{\\left(\\frac{2}{3}\\right)}^{2}=\\frac{4}{9}.\\,[\/latex]Therefore,[latex]\\,{\\mathrm{log}}_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2.[\/latex]<\/li>\n<\/ul>\n<div id=\"fs-id1165137455840\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137453770\"><strong>Given a logarithm of the form[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right),[\/latex]evaluate it mentally.<\/strong><\/p>\n<ol id=\"fs-id1165134079724\" type=\"1\">\n<li>Rewrite the argument[latex]\\,x\\,[\/latex]as a power of[latex]\\,b:\\,[\/latex][latex]{b}^{y}=x.\\,[\/latex]<\/li>\n<li>Use previous knowledge of powers of[latex]\\,b\\,[\/latex]identify[latex]\\,y\\,[\/latex]by asking, \u201cTo what exponent should[latex]\\,b\\,[\/latex]be raised in order to get[latex]\\,x?[\/latex]\u201d<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137732842\">\n<div id=\"fs-id1165135296345\">\n<h3>Solving Logarithms Mentally<\/h3>\n<p id=\"fs-id1165135393440\">Solve[latex]\\,y={\\mathrm{log}}_{4}\\left(64\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1165137852123\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137611276\">First we rewrite the logarithm in exponential form:[latex]\\,{4}^{y}=64.\\,[\/latex]Next, we ask, \u201cTo what exponent must 4 be raised in order to get 64?\u201d<\/p>\n<p>We know<\/p>\n<div id=\"eip-id1165134583995\" class=\"unnumbered\">[latex]{4}^{3}=64[\/latex]<\/div>\n<p id=\"fs-id1165137619013\">Therefore,<\/p>\n<div id=\"eip-id1165135606935\" class=\"unnumbered\">[latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731430\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_03\">\n<div id=\"fs-id1165137704553\">\n<p id=\"fs-id1165137745041\">Solve[latex]\\,y={\\mathrm{log}}_{121}\\left(11\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1165137693554\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137639199\">[latex]{\\mathrm{log}}_{121}\\left(11\\right)=\\frac{1}{2}\\,[\/latex](recalling that[latex]\\,\\sqrt{121}={\\left(121\\right)}^{\\frac{1}{2}}=11[\/latex])<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137663658\">\n<div id=\"fs-id1165137680390\">\n<h3>Evaluating the Logarithm of a Reciprocal<\/h3>\n<p id=\"fs-id1165137938805\">Evaluate[latex]\\,y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1165135526087\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137638179\">First we rewrite the logarithm in exponential form:[latex]\\,{3}^{y}=\\frac{1}{27}.\\,[\/latex]Next, we ask, \u201cTo what exponent must 3 be raised in order to get[latex]\\,\\frac{1}{27}?[\/latex]\u201d<\/p>\n<p id=\"fs-id1165137552085\">We know[latex]\\,{3}^{3}=27,[\/latex]but what must we do to get the reciprocal,[latex]\\,\\frac{1}{27}?\\,[\/latex]Recall from working with exponents that[latex]\\,{b}^{-a}=\\frac{1}{{b}^{a}}.\\,[\/latex]We use this information to write<\/p>\n<div id=\"eip-id1165137550550\" class=\"unnumbered\">[latex]\\begin{array}{l}{3}^{-3}=\\frac{1}{{3}^{3}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{1}{27}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137585807\">Therefore,[latex]\\,{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575754\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_04\">\n<div id=\"fs-id1165137768727\">\n<p id=\"fs-id1165135437134\">Evaluate[latex]\\,y={\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1165137812401\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137806792\">[latex]{\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)=-5[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137547253\" class=\"bc-section section\">\n<h3>Using Common Logarithms<\/h3>\n<p id=\"fs-id1165137574205\">Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression[latex]\\,\\mathrm{log}\\left(x\\right)\\,[\/latex]means[latex]\\,{\\mathrm{log}}_{10}\\left(x\\right).\\,[\/latex]We call a base-10 logarithm a <strong>common logarithm<\/strong>. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.<\/p>\n<div id=\"fs-id1165137401037\" class=\"textbox key-takeaways\">\n<h3>Definition of the Common Logarithm<\/h3>\n<p id=\"fs-id1165135609332\">A common logarithm is a logarithm with base[latex]\\,10.\\,[\/latex]We write[latex]\\,{\\mathrm{log}}_{10}\\left(x\\right)\\,[\/latex]simply as[latex]\\,\\mathrm{log}\\left(x\\right).\\,[\/latex]The common logarithm of a positive number[latex]\\,x\\,[\/latex]satisfies the following definition.<\/p>\n<p id=\"fs-id1165137601579\">For[latex]\\,x>0,[\/latex]<\/p>\n<div id=\"fs-id1165137475905\">[latex]y=\\mathrm{log}\\left(x\\right)\\text{ is equivalent to }{10}^{y}=x[\/latex]<\/div>\n<p id=\"fs-id1165137559681\">We read[latex]\\,\\mathrm{log}\\left(x\\right)\\,[\/latex]as, \u201cthe logarithm with base[latex]\\,10\\,[\/latex]of[latex]\\,x\\,[\/latex]\u201d or \u201clog base 10 of[latex]\\,x.[\/latex]\u201d<\/p>\n<p id=\"fs-id1165137771789\">The logarithm[latex]\\,y\\,[\/latex]is the exponent to which[latex]\\,10\\,[\/latex]must be raised to get[latex]\\,x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137579434\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137810781\"><strong>Given a common logarithm of the form[latex]\\,y=\\mathrm{log}\\left(x\\right),[\/latex] evaluate it mentally.<\/strong><\/p>\n<ol id=\"fs-id1165137828334\" type=\"1\">\n<li>Rewrite the argument[latex]\\,x\\,[\/latex]as a power of[latex]\\,10:\\,[\/latex][latex]{10}^{y}=x.[\/latex]<\/li>\n<li>Use previous knowledge of powers of[latex]\\,10\\,[\/latex]to identify[latex]\\,y\\,[\/latex]by asking, \u201cTo what exponent must[latex]\\,10\\,[\/latex]be raised in order to get[latex]\\,x?[\/latex]\u201d<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137742366\">\n<div id=\"fs-id1165137418239\">\n<h3>Finding the Value of a Common Logarithm Mentally<\/h3>\n<p id=\"fs-id1165137658546\">Evaluate[latex]\\,y=\\mathrm{log}\\left(1000\\right)\\,[\/latex]without using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1165137634154\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137444192\">First we rewrite the logarithm in exponential form:[latex]\\,{10}^{y}=1000.\\,[\/latex]Next, we ask, \u201cTo what exponent must[latex]\\,10\\,[\/latex]be raised in order to get 1000?\u201d We know<\/p>\n<div id=\"eip-id1165134331119\" class=\"unnumbered\">[latex]{10}^{3}=1000[\/latex]<\/div>\n<p id=\"fs-id1165137584125\">Therefore,[latex]\\,\\mathrm{log}\\left(1000\\right)=3.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503827\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_05\">\n<div id=\"fs-id1165137673696\">\n<p id=\"fs-id1165137393877\">Evaluate[latex]\\,y=\\mathrm{log}\\left(1,000,000\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137768485\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137436094\">[latex]\\mathrm{log}\\left(1,000,000\\right)=6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137552804\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137827812\"><strong>Given a common logarithm with the form[latex]\\,y=\\mathrm{log}\\left(x\\right),[\/latex]evaluate it using a calculator.<\/strong><\/p>\n<ol id=\"fs-id1165137418685\" type=\"1\">\n<li>Press <strong>[LOG]<\/strong>.<\/li>\n<li>Enter the value given for[latex]\\,x,[\/latex]followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137793928\">\n<div id=\"fs-id1165137892249\">\n<h3>Finding the Value of a Common Logarithm Using a Calculator<\/h3>\n<p id=\"fs-id1165137667877\">Evaluate[latex]\\,y=\\mathrm{log}\\left(321\\right)\\,[\/latex]to four decimal places using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1165137404714\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ul id=\"fs-id1165137786486\">\n<li>Press <strong>[LOG]<\/strong>.<\/li>\n<li>Enter 321<em>,<\/em> followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137735413\">Rounding to four decimal places,[latex]\\,\\mathrm{log}\\left(321\\right)\\approx 2.5065.[\/latex]<\/p>\n<\/details>\n<\/div>\n<div>\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137789015\">Note that[latex]\\,{10}^{2}=100\\,[\/latex]and that[latex]\\,{10}^{3}=1000.\\,[\/latex]Since 321 is between 100 and 1000, we know that[latex]\\,\\mathrm{log}\\left(321\\right)\\,[\/latex]must be between[latex]\\,\\mathrm{log}\\left(100\\right)\\,[\/latex]and[latex]\\,\\mathrm{log}\\left(1000\\right).\\,[\/latex]This gives us the following:<\/p>\n<div id=\"eip-id1165134280435\" class=\"unnumbered\">[latex]\\begin{array}{ccccc}100& <& 321& <& 1000\\\\ 2& <& 2.5065& <& 3\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137780842\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_06\">\n<div id=\"fs-id1165135241210\">\n<p id=\"fs-id1165137735373\">Evaluate[latex]\\,y=\\mathrm{log}\\left(123\\right)\\,[\/latex]to four decimal places using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1165137550190\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137844052\">[latex]\\mathrm{log}\\left(123\\right)\\approx 2.0899[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_04_03_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137603561\">\n<div id=\"fs-id1165135704023\">\n<h3>Rewriting and Solving a Real-World Exponential Model<\/h3>\n<p id=\"fs-id1165135194300\">The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation[latex]\\,{10}^{x}=500\\,[\/latex]represents this situation, where[latex]\\,x\\,[\/latex]is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?<\/p>\n<\/div>\n<div id=\"fs-id1165137784516\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137827621\">We begin by rewriting the exponential equation in logarithmic form.<\/p>\n<div id=\"eip-id1165134048114\" class=\"unnumbered\">[latex]\\begin{array}{lll}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{10}^{x}\\hfill & =500\\hfill & \\hfill \\\\ \\mathrm{log}\\left(500\\right)\\hfill & =x\\hfill & \\text{Use the definition of the common log}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137419444\">Next we evaluate the logarithm using a calculator:<\/p>\n<ul id=\"fs-id1165137736356\">\n<li>Press <strong>[LOG]<\/strong>.<\/li>\n<li>Enter[latex]\\,500,[\/latex]followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<li>To the nearest thousandth,[latex]\\,\\mathrm{log}\\left(500\\right)\\approx 2.699.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137422793\">The difference in magnitudes was about[latex]\\,2.699.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137749635\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_07\">\n<div id=\"fs-id1165135195254\">\n<p id=\"fs-id1165137736970\">The amount of energy released from one earthquake was[latex]\\,\\text{8,500}\\,[\/latex]times greater than the amount of energy released from another. The equation[latex]\\,{10}^{x}=8500\\,[\/latex]represents this situation, where[latex]\\,x\\,[\/latex]is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?<\/p>\n<\/div>\n<div id=\"fs-id1165137656499\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137438675\">The difference in magnitudes was about [latex]\\,3.929.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137405741\" class=\"bc-section section\">\n<h3>Using Natural Logarithms<\/h3>\n<p id=\"fs-id1165137661970\">The most frequently used base for logarithms is[latex]\\,e.\\,[\/latex]Base[latex]\\,e\\,[\/latex]logarithms are important in calculus and some scientific applications; they are called <strong>natural logarithms<\/strong>. The base[latex]\\,e\\,[\/latex]logarithm,[latex]\\,{\\mathrm{log}}_{e}\\left(x\\right),[\/latex] has its own notation,[latex]\\,\\mathrm{ln}\\left(x\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137473872\">Most values of[latex]\\,\\mathrm{ln}\\left(x\\right)\\,[\/latex]can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,[latex]\\,\\mathrm{ln}1=0.\\,[\/latex]For other natural logarithms, we can use the[latex]\\,\\mathrm{ln}\\,[\/latex]key that can be found on most scientific calculators. We can also find the natural logarithm of any power of[latex]\\,e\\,[\/latex]using the inverse property of logarithms.<\/p>\n<div id=\"fs-id1165137452317\" class=\"textbox key-takeaways\">\n<h3>Definition of the Natural Logarithm<\/h3>\n<p id=\"fs-id1165137579241\">A natural logarithm is a logarithm with base[latex]\\,e.[\/latex] We write [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] simply as [latex]\\mathrm{ln}\\left(x\\right).[\/latex] The natural logarithm of a positive number [latex]x[\/latex] satisfies the following definition.<\/p>\n<p id=\"fs-id1165135613642\">For[latex]\\,x>0,[\/latex]<\/p>\n<div id=\"fs-id1165137580230\">[latex]y=\\mathrm{ln}\\left(x\\right)\\text{ is equivalent to }{e}^{y}=x[\/latex]<\/div>\n<p id=\"fs-id1165137658264\">We read[latex]\\,\\mathrm{ln}\\left(x\\right)\\,[\/latex]as, \u201cthe logarithm with base[latex]\\,e\\,[\/latex]of[latex]\\,x[\/latex]\u201d or \u201cthe natural logarithm of[latex]\\,x.[\/latex]\u201d<\/p>\n<p id=\"fs-id1165137566720\">The logarithm[latex]\\,y\\,[\/latex]is the exponent to which[latex]\\,e\\,[\/latex]must be raised to get[latex]\\,x.[\/latex]<\/p>\n<p id=\"fs-id1165137705251\">Since the functions[latex]\\,y=e{}^{x}\\,[\/latex]and[latex]\\,y=\\mathrm{ln}\\left(x\\right)\\,[\/latex]are inverse functions,[latex]\\,\\mathrm{ln}\\left({e}^{x}\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]and[latex]\\,e{}^{\\mathrm{ln}\\left(x\\right)}=x\\,[\/latex]for[latex]\\,x>0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137409558\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137832169\"><strong>Given a natural logarithm with the form[latex]\\,y=\\mathrm{ln}\\left(x\\right),[\/latex] evaluate it using a calculator.<\/strong><\/p>\n<ol id=\"fs-id1165135407195\" type=\"1\">\n<li>Press <strong>[LN]<\/strong>.<\/li>\n<li>Enter the value given for[latex]\\,x,[\/latex] followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_03_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137731536\">\n<div id=\"fs-id1165137434974\">\n<h3>Evaluating a Natural Logarithm Using a Calculator<\/h3>\n<p id=\"fs-id1165137573341\">Evaluate[latex]\\,y=\\mathrm{ln}\\left(500\\right)\\,[\/latex]to four decimal places using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1165137702133\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ul id=\"fs-id1165137563770\">\n<li>Press <strong>[LN]<\/strong>.<\/li>\n<li>Enter[latex]\\,500,[\/latex]followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137645024\">Rounding to four decimal places,[latex]\\,\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137676028\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_08\">\n<div id=\"fs-id1165137431140\">\n<p id=\"fs-id1165137435623\">Evaluate[latex]\\,\\mathrm{ln}\\left(-500\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137737001\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137639598\">It is not possible to take the logarithm of a negative number in the set of real numbers.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137648012\" class=\"precalculus media\">\n<p id=\"fs-id1165137451079\">Access this online resource for additional instruction and practice with logarithms.<\/p>\n<ul id=\"fs-id1165137732886\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/intrologarithms\">Introduction to Logarithms<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137870892\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id1983134\" summary=\"...\">\n<tbody>\n<tr>\n<td>Definition of the logarithmic function<\/td>\n<td>For[latex]\\text{ } x>0,b>0,b\\ne 1,[\/latex]&lt;[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\text{ }[\/latex]if and only if[latex]\\text{ }{b}^{y}=x.[\/latex]&lt;\/td&gt;<\/td>\n<\/tr>\n<tr>\n<td>Definition of the common logarithm<\/td>\n<td>For[latex]\\text{ }x>0,[\/latex][latex]y=\\mathrm{log}\\left(x\\right)\\text{ }[\/latex]if and only if[latex]\\text{ }{10}^{y}=x.[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Definition of the natural logarithm<\/td>\n<td>For[latex]\\text{ }x>0,[\/latex][latex]y=\\mathrm{ln}\\left(x\\right)\\text{ }[\/latex]if and only if[latex]\\text{ }{e}^{y}=x.[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135699130\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137574258\">\n<li>The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.<\/li>\n<li>Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See <a class=\"autogenerated-content\" href=\"#Example_04_03_01\">(Figure)<\/a>.<\/li>\n<li>Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See <a class=\"autogenerated-content\" href=\"#Example_04_03_02\">(Figure)<\/a>.<\/li>\n<li>Logarithmic functions with base[latex]\\,b\\,[\/latex]can be evaluated mentally using previous knowledge of powers of[latex]\\,b.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_03_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_04_03_04\">(Figure)<\/a>.<\/li>\n<li>Common logarithms can be evaluated mentally using previous knowledge of powers of[latex]\\,10.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_03_05\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>When common logarithms cannot be evaluated mentally, a calculator can be used. See <a class=\"autogenerated-content\" href=\"#Example_04_03_06\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>Real-world exponential problems with base[latex]\\,10\\,[\/latex]can be rewritten as a common logarithm and then evaluated using a calculator. See <a class=\"autogenerated-content\" href=\"#Example_04_03_07\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>Natural logarithms can be evaluated using a calculator <a class=\"autogenerated-content\" href=\"#Example_04_03_08\">(Figure)<\/a><strong>.<\/strong><\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135192789\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137427076\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137817361\">\n<div id=\"fs-id1165137559978\">\n<p id=\"fs-id1165137445179\">What is a base[latex]\\,b\\,[\/latex]logarithm? Discuss the meaning by interpreting each part of the equivalent equations[latex]\\,{b}^{y}=x\\,[\/latex]and[latex]\\,{\\mathrm{log}}_{b}x=y\\,[\/latex]for[latex]\\,b>0,b\\ne 1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137411524\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>A logarithm is an exponent. Specifically, it is the exponent to which a base[latex]\\,b\\,[\/latex]is raised to produce a given value. In the expressions given, the base[latex]\\,b\\,[\/latex]has the same value. The exponent,[latex]\\,y,[\/latex]in the expression[latex]\\,{b}^{y}\\,[\/latex]can also be written as the logarithm,[latex]\\,{\\mathrm{log}}_{b}x,[\/latex]and the value of[latex]\\,x\\,[\/latex]is the result of raising[latex]\\,b\\,[\/latex]to the power of[latex]\\,y.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137574896\">\n<div id=\"fs-id1165137658231\">\n<p id=\"fs-id1165137529141\">How is the logarithmic function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}x\\,[\/latex]related to the exponential function[latex]\\,g\\left(x\\right)={b}^{x}?\\,[\/latex]What is the result of composing these two functions?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137446568\">\n<div id=\"fs-id1165137602953\">\n<p id=\"fs-id1165137557324\">How can the logarithmic equation[latex]\\,{\\mathrm{log}}_{b}x=y\\,[\/latex]be solved for[latex]\\,x\\,[\/latex]using the properties of exponents?<\/p>\n<\/div>\n<div id=\"fs-id1165135203815\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137530592\">Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation[latex]\\,{b}^{y}=x,[\/latex] and then properties of exponents can be applied to solve for[latex]\\,x.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137470358\">\n<div id=\"fs-id1165135526986\">\n<p id=\"fs-id1165137697077\">Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base[latex]\\,b,[\/latex] and how does the notation differ?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137507578\">\n<div id=\"fs-id1165137828407\">\n<p id=\"fs-id1165135417853\">Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base[latex]\\,b,[\/latex] and how does the notation differ?<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137596568\">The natural logarithm is a special case of the logarithm with base[latex]\\,b\\,[\/latex]in that the natural log always has base[latex]\\,e.\\,[\/latex]Rather than notating the natural logarithm as[latex]\\,{\\mathrm{log}}_{e}\\left(x\\right),[\/latex]the notation used is[latex]\\,\\mathrm{ln}\\left(x\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447239\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137414571\">For the following exercises, rewrite each equation in exponential form.<\/p>\n<div id=\"fs-id1165137646887\">\n<div id=\"fs-id1165137664870\">\n<p id=\"fs-id1165137726644\">[latex]{\\text{log}}_{4}\\left(q\\right)=m[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137473551\">\n<div id=\"fs-id1165137454663\">\n<p id=\"fs-id1165137736589\">[latex]{\\text{log}}_{a}\\left(b\\right)=c[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135191877\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137531424\">[latex]{a}^{c}=b[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137506749\">\n<div id=\"fs-id1165135536326\">\n<p id=\"fs-id1165137430583\">[latex]{\\mathrm{log}}_{16}\\left(y\\right)=x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410961\">\n<div id=\"fs-id1165137602136\">\n<p id=\"fs-id1165137654948\">[latex]{\\mathrm{log}}_{x}\\left(64\\right)=y[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137428120\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135192141\">[latex]{x}^{y}=64[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673422\">\n<div id=\"fs-id1165135585639\">\n<p id=\"fs-id1165135516961\">[latex]{\\mathrm{log}}_{y}\\left(x\\right)=-11[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135356538\">\n<div id=\"fs-id1165137501385\">\n<p id=\"fs-id1165137629520\">[latex]{\\mathrm{log}}_{15}\\left(a\\right)=b[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135191871\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]{15}^{b}=a[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137580855\">\n<div id=\"fs-id1165137438413\">\n<p id=\"fs-id1165137656672\">[latex]{\\mathrm{log}}_{y}\\left(137\\right)=x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137611652\">\n<div id=\"fs-id1165134038222\">\n<p id=\"fs-id1165137645621\">[latex]{\\mathrm{log}}_{13}\\left(142\\right)=a[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137580134\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135174998\">[latex]{13}^{a}=142[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137724172\">\n<div id=\"fs-id1165137724174\">\n<p id=\"fs-id1165134077330\">[latex]\\text{log}\\left(v\\right)=t[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735890\">\n<div id=\"fs-id1165137735892\">\n<p id=\"fs-id1165137560511\">[latex]\\text{ln}\\left(w\\right)=n[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135185210\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135185212\">[latex]{e}^{n}=w[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137810335\">For the following exercises, rewrite each equation in logarithmic form.<\/p>\n<div id=\"fs-id1165135186264\">\n<div id=\"fs-id1165135203858\">\n<p id=\"fs-id1165135203860\">[latex]{4}^{x}=y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135194695\">\n<div id=\"fs-id1165137724816\">\n<p id=\"fs-id1165137724818\">[latex]{c}^{d}=k[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137742265\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137742268\">[latex]{\\text{log}}_{c}\\left(k\\right)=d[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137698394\">\n<div id=\"fs-id1165137434093\">\n<p id=\"fs-id1165137434095\">[latex]{m}^{-7}=n[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135384408\">\n<div id=\"fs-id1165135384410\">\n<p id=\"fs-id1165137534920\">[latex]{19}^{x}=y[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135296129\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137771226\">[latex]{\\mathrm{log}}_{19}y=x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135358961\">\n<div id=\"fs-id1165137627327\">\n<p id=\"fs-id1165137911046\">[latex]{x}^{-\\,\\frac{10}{13}}=y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132971714\">\n<div id=\"fs-id1165132971716\">\n<p id=\"fs-id1165137728409\">[latex]{n}^{4}=103[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134032258\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137447080\">[latex]{\\mathrm{log}}_{n}\\left(103\\right)=4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137564489\">\n<div id=\"fs-id1165137564491\">\n<p id=\"fs-id1165137404398\">[latex]{\\left(\\frac{7}{5}\\right)}^{m}=n[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731983\">\n<div id=\"fs-id1165137597971\">\n<p id=\"fs-id1165137597973\">[latex]{y}^{x}=\\frac{39}{100}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135194466\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135194469\">[latex]{\\mathrm{log}}_{y}\\left(\\frac{39}{100}\\right)=x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137827739\">\n<div id=\"fs-id1165137762434\">\n<p id=\"fs-id1165137762436\">[latex]{10}^{a}=b[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135296148\">\n<div id=\"fs-id1165134040600\">\n<p id=\"fs-id1165134040602\">[latex]{e}^{k}=h[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135191801\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135191803\">[latex]\\text{ln}\\left(h\\right)=k[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137762947\">For the following exercises, solve for[latex]\\,x\\,[\/latex]by converting the logarithmic equation to exponential form.<\/p>\n<div>\n<div id=\"fs-id1165137438779\">\n<p id=\"fs-id1165137823075\">[latex]{\\text{log}}_{3}\\left(x\\right)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137482919\">\n<div id=\"fs-id1165137628211\">\n<p id=\"fs-id1165137628213\">[latex]{\\text{log}}_{2}\\left(x\\right)=-3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137443266\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137834752\">[latex]x={2}^{-3}=\\frac{1}{8}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137726345\">\n<div id=\"fs-id1165137726348\">\n<p id=\"fs-id1165137473443\">[latex]{\\text{log}}_{5}\\left(x\\right)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137389073\">\n<div id=\"fs-id1165137639044\">\n<p id=\"fs-id1165137639046\">[latex]{\\mathrm{log}}_{3}\\left(x\\right)=3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137409760\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137409762\">[latex]x={3}^{3}=27[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661950\">\n<div id=\"fs-id1165137726952\">\n<p id=\"fs-id1165137475701\">[latex]{\\text{log}}_{2}\\left(x\\right)=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627690\">\n<div>\n<p id=\"fs-id1165137444979\">[latex]{\\text{log}}_{9}\\left(x\\right)=\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137469004\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137651832\">[latex]x={9}^{\\frac{1}{2}}=3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203739\">\n<div>\n<p id=\"fs-id1165137427716\">[latex]{\\text{log}}_{18}\\left(x\\right)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613168\">\n<div id=\"fs-id1165137414109\">\n<p id=\"fs-id1165137414111\">[latex]{\\mathrm{log}}_{6}\\left(x\\right)=-3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135253754\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137543450\">[latex]x={6}^{-3}=\\frac{1}{216}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137937666\">\n<div id=\"fs-id1165137937668\">\n<p id=\"fs-id1165137758189\">[latex]\\text{log}\\left(x\\right)=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736080\">\n<div id=\"fs-id1165137442469\">\n<p id=\"fs-id1165137442471\">[latex]\\text{ln}\\left(x\\right)=2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135524549\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135524551\">[latex]x={e}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137634430\">For the following exercises, use the definition of common and natural logarithms to simplify.<\/p>\n<div id=\"fs-id1165137634434\">\n<div id=\"fs-id1165137732545\">\n<p id=\"fs-id1165137732547\">[latex]\\text{log}\\left({100}^{8}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137936925\">\n<div id=\"fs-id1165137936927\">\n<p id=\"fs-id1165135526105\">[latex]{10}^{\\text{log}\\left(32\\right)}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137436613\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137942473\">[latex]32[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137529000\">\n<div id=\"fs-id1165137529002\">\n<p id=\"fs-id1165135152291\">[latex]2\\text{log}\\left(.0001\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134079645\">\n<div id=\"fs-id1165137566210\">\n<p id=\"fs-id1165137566212\">[latex]{e}^{\\mathrm{ln}\\left(1.06\\right)}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137455559\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137464570\">[latex]1.06[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137400162\">\n<div id=\"fs-id1165137400164\">\n<p id=\"fs-id1165135259620\">[latex]\\mathrm{ln}\\left({e}^{-5.03}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137543862\">\n<div id=\"fs-id1165137543865\">\n<p id=\"fs-id1165135434891\">[latex]{e}^{\\mathrm{ln}\\left(10.125\\right)}+4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137605063\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]14.125[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451809\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137564770\">For the following exercises, evaluate the base[latex]\\,b\\,[\/latex]logarithmic expression without using a calculator.<\/p>\n<div id=\"fs-id1165137676437\">\n<div id=\"fs-id1165137676439\">\n<p id=\"fs-id1165137728122\">[latex]{\\text{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137590077\">\n<div id=\"fs-id1165137590079\">\n<p id=\"fs-id1165134108525\">[latex]{\\text{log}}_{6}\\left(\\sqrt{6}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135706822\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137911553\">[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137415151\">\n<div id=\"fs-id1165137423807\">\n<p id=\"fs-id1165137423810\">[latex]{\\text{log}}_{2}\\left(\\frac{1}{8}\\right)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190140\">\n<div id=\"fs-id1165137806053\">\n<p id=\"fs-id1165137806055\">[latex]6{\\text{log}}_{8}\\left(4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137454533\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137454536\">[latex]4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137532658\">For the following exercises, evaluate the common logarithmic expression without using a calculator.<\/p>\n<div id=\"fs-id1165137542512\">\n<div id=\"fs-id1165135613435\">\n<p id=\"fs-id1165135613437\">[latex]\\text{log}\\left(10,000\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137755960\">\n<div id=\"fs-id1165137736340\">\n<p id=\"fs-id1165137736342\">[latex]\\text{log}\\left(0.001\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135431079\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135431081\">[latex]-\\text{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190197\">\n<div id=\"fs-id1165135190199\">\n<p id=\"fs-id1165137422196\">[latex]\\text{log}\\left(1\\right)+7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137432736\">\n<div id=\"fs-id1165137655695\">\n<p id=\"fs-id1165137655697\">[latex]2\\text{log}\\left({100}^{-3}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137588216\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137588218\">[latex]-12[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137570442\">For the following exercises, evaluate the natural logarithmic expression without using a calculator.<\/p>\n<div id=\"fs-id1165137619029\">\n<div id=\"fs-id1165137619032\">\n<p id=\"fs-id1165134047613\">[latex]\\text{ln}\\left({e}^{\\frac{1}{3}}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736480\">\n<div id=\"fs-id1165137736482\">\n<p id=\"fs-id1165137645639\">[latex]\\text{ln}\\left(1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137645466\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135207524\">[latex]0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137804477\">\n<div id=\"fs-id1165137474568\">\n<p id=\"fs-id1165137474570\">[latex]\\text{ln}\\left({e}^{-0.225}\\right)-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137762957\">\n<div id=\"fs-id1165135570229\">\n<p id=\"fs-id1165135570231\">[latex]25\\text{ln}\\left({e}^{\\frac{2}{5}}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137418633\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137418635\">[latex]10[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137527430\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137401108\">For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.<\/p>\n<div id=\"fs-id1165137626617\">\n<div id=\"fs-id1165137626619\">\n<p id=\"fs-id1165137793717\">[latex]\\text{log}\\left(0.04\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640432\">\n<div id=\"fs-id1165137640434\">\n<p id=\"fs-id1165135194090\">[latex]\\text{ln}\\left(15\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137696603\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137679201\">[latex]\\text{2}.\\text{7}0\\text{8}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731720\">\n<div id=\"fs-id1165137731722\">\n<p id=\"fs-id1165137463176\">[latex]\\text{ln}\\left(\\frac{4}{5}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137527828\">\n<div id=\"fs-id1165137527830\">\n<p id=\"fs-id1165137657921\">[latex]\\text{log}\\left(\\sqrt{2}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137658294\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137405172\">[latex]0.151[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137655087\">\n<div id=\"fs-id1165137655089\">\n<p id=\"fs-id1165137757908\">[latex]\\text{ln}\\left(\\sqrt{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932472\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137435099\">\n<div id=\"fs-id1165137593336\">\n<p id=\"fs-id1165137593338\">Is[latex]\\,x=0\\,[\/latex]in the domain of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(x\\right)?\\,[\/latex]If so, what is the value of the function when[latex]\\,x=0?\\,[\/latex]Verify the result.<\/p>\n<\/div>\n<div id=\"fs-id1165137474123\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137911451\">No, the function has no defined value for[latex]\\,x=0.\\,[\/latex]To verify, suppose[latex]\\,x=0\\,[\/latex]is in the domain of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(x\\right).\\,[\/latex]Then there is some number[latex]\\,n\\,[\/latex]such that[latex]\\,n=\\mathrm{log}\\left(0\\right).\\,[\/latex]Rewriting as an exponential equation gives:[latex]\\,{10}^{n}=0,[\/latex] which is impossible since no such real number[latex]\\,n\\,[\/latex]exists. Therefore,[latex]\\,x=0\\,[\/latex]is <em>not<\/em> the domain of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(x\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137597455\">\n<div id=\"fs-id1165137597458\">\n<p id=\"fs-id1165137627827\">Is[latex]\\,f\\left(x\\right)=0\\,[\/latex]in the range of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(x\\right)?\\,[\/latex]If so, for what value of[latex]\\,x?\\,[\/latex]Verify the result.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137548869\">\n<div id=\"fs-id1165137548871\">\n<p id=\"fs-id1165135613701\">Is there a number[latex]\\,x\\,[\/latex]such that[latex]\\,\\mathrm{ln}x=2?\\,[\/latex]If so, what is that number? Verify the result.<\/p>\n<\/div>\n<div id=\"fs-id1165137659786\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137806408\">Yes. Suppose there exists a real number[latex]\\,x\\,[\/latex]such that[latex]\\,\\mathrm{ln}x=2.\\,[\/latex]Rewriting as an exponential equation gives[latex]\\,x={e}^{2},[\/latex] which is a real number. To verify, let[latex]\\,x={e}^{2}.\\,[\/latex]Then, by definition,[latex]\\,\\mathrm{ln}\\left(x\\right)=\\mathrm{ln}\\left({e}^{2}\\right)=2.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640809\">\n<div id=\"fs-id1165137603717\">\n<p id=\"fs-id1165137603719\">Is the following true:[latex]\\,\\frac{{\\mathrm{log}}_{3}\\left(27\\right)}{{\\mathrm{log}}_{4}\\left(\\frac{1}{64}\\right)}=-1?\\,[\/latex]Verify the result.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137453751\">\n<div id=\"fs-id1165137453754\">\n<p id=\"fs-id1165137413735\">Is the following true:[latex]\\,\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}=1.725?\\,[\/latex]Verify the result.<\/p>\n<\/div>\n<div id=\"fs-id1165137749907\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137559774\">No;[latex]\\,\\mathrm{ln}\\left(1\\right)=0,[\/latex] so[latex]\\,\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}\\,[\/latex]is undefined.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135437184\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165137732791\">\n<div id=\"fs-id1165137913993\">\n<p id=\"fs-id1165137913995\">The exposure index[latex]\\,EI\\,[\/latex]for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation[latex]\\,EI={\\mathrm{log}}_{2}\\left(\\frac{{f}^{2}}{t}\\right),[\/latex] where[latex]\\,f\\,[\/latex]is the \u201cf-stop\u201d setting on the camera, and [latex]t[\/latex] is the exposure time in seconds. Suppose the f-stop setting is[latex]\\,8\\,[\/latex]and the desired exposure time is[latex]\\,2\\,[\/latex]seconds. What will the resulting exposure index be?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137602359\">\n<div id=\"fs-id1165137838750\">\n<p id=\"fs-id1165137838752\">Refer to the previous exercise. Suppose the light meter on a camera indicates an[latex]\\,EI\\,[\/latex]of[latex]\\,-2,[\/latex] and the desired exposure time is 16 seconds. What should the f-stop setting be?<\/p>\n<\/div>\n<div id=\"fs-id1165137570610\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137417561\">[latex]2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137697043\">The intensity levels <em>I<\/em> of two earthquakes measured on a seismograph can be compared by the formula[latex]\\,\\mathrm{log}\\frac{{I}_{1}}{{I}_{2}}={M}_{1}-{M}_{2}\\,[\/latex]where[latex]\\,M\\,[\/latex]is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/world\/historical.php. Accessed 3\/4\/2014.\" id=\"return-footnote-102-5\" href=\"#footnote-102-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a> How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135160066\">\n<dt>common logarithm<\/dt>\n<dd id=\"fs-id1165137571387\">the exponent to which 10 must be raised to get[latex]\\,x;\\,[\/latex][latex]\\,{\\mathrm{log}}_{10}\\left(x\\right)\\,[\/latex] is written simply as[latex]\\,\\mathrm{log}\\left(x\\right).[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137780762\">\n<dt>logarithm<\/dt>\n<dd id=\"fs-id1165137849198\">the exponent to which[latex]\\,b\\,[\/latex]must be raised to get[latex]\\,x;\\,[\/latex]written[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl>\n<dt>natural logarithm<\/dt>\n<dd>the exponent to which the number[latex]\\,e\\,[\/latex]must be raised to get[latex]\\,x;[\/latex][latex]\\,{\\mathrm{log}}_{e}\\left(x\\right)\\,[\/latex]is written as[latex]\\,\\mathrm{ln}\\left(x\\right).[\/latex]<\/dd>\n<\/dl>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-102-1\"><a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary<\/a>. Accessed 3\/4\/2013. <a href=\"#return-footnote-102-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-102-2\"><a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary<\/a>. Accessed 3\/4\/2013. <a href=\"#return-footnote-102-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-102-3\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/. Accessed 3\/4\/2013. <a href=\"#return-footnote-102-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-102-4\"><a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/<\/a>. Accessed 3\/4\/2013. <a href=\"#return-footnote-102-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-102-5\"><a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/world\/historical.php\">http:\/\/earthquake.usgs.gov\/earthquakes\/world\/historical.php<\/a>. Accessed 3\/4\/2014. <a href=\"#return-footnote-102-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":291,"menu_order":4,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-102","chapter","type-chapter","status-publish","hentry"],"part":95,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/102","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\/102\/revisions"}],"predecessor-version":[{"id":103,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/102\/revisions\/103"}],"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\/102\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=102"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=102"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=102"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}