{"id":221,"date":"2023-11-08T16:10:30","date_gmt":"2023-11-08T16:10:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/convert-from-logarithmic-to-exponential-form\/"},"modified":"2024-08-01T00:59:01","modified_gmt":"2024-08-01T00:59:01","slug":"7-4-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/7-4-logarithmic-functions\/","title":{"raw":"7.4 Logarithmic Functions","rendered":"7.4 Logarithmic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate logarithmic expressions using the definition of logarithms.<\/li>\r\n \t<li>Graph logarithmic functions by creating a table of values.<\/li>\r\n \t<li>Find the equation of the asymptote for a logarithmic function.<\/li>\r\n \t<li>Find the domain and range of a logarithmic function using its graph and write them in interval notation.<\/li>\r\n \t<li>Rewrite exponential equations in logarithmic form.<\/li>\r\n \t<li>Rewrite logarithmic equations in exponential form.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<figure id=\"CNX_Precalc_Figure_04_03_001\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051922\/CNX_Precalc_Figure_04_03_0012.jpg\" alt=\"Photo of the aftermath of the earthquake in Japan with a focus on the Japanese flag.\" width=\"488\" height=\"325\" \/> Devastation of the earthquake which happened on March 11, 2011 in Honshu, Japan. (credit: Daniel Pierce)[\/caption]<\/figure>\r\n<p id=\"fs-id1165137557013\">In [latex]2010[\/latex], a major earthquake struck Haiti, destroying or damaging over\u00a0[latex]285,000[\/latex] homes.[footnote]<a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary\" target=\"_blank\" rel=\"noopener\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary<\/a>. Accessed\u00a0[latex]3\/4\/2013[\/latex].[\/footnote] One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over\u00a0[latex]332,000[\/latex] buildings,[footnote]<a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary\" target=\"_blank\" rel=\"noopener\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary<\/a>. Accessed\u00a0[latex]3\/4\/2013[\/latex].[\/footnote]\u00a0like those shown in the picture above. Even though both caused substantial damage, the earthquake in\u00a0[latex]2011[\/latex] was\u00a0[latex]100[\/latex] 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\u00a0[latex]7.0[\/latex] on the Richter Scale[footnote]<a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/\" target=\"_blank\" rel=\"noopener\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/<\/a>. Accessed\u00a0[latex]3\/4\/2013[\/latex].[\/footnote]\u00a0whereas the Japanese earthquake registered a\u00a0[latex]9.0[\/latex].[footnote]<a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#details\" target=\"_blank\" rel=\"noopener\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#details<\/a>. Accessed\u00a0[latex]3\/4\/2013[\/latex].[\/footnote]<\/p>\r\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\u00a0[latex]4[\/latex]. 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>\r\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\u00a0[latex]500[\/latex] 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 <strong>Richter Scale<\/strong>. How would we solve for [latex]x[\/latex]?<\/p>\r\nWe 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]\u00a0must be some value between\u00a0[latex]2[\/latex] and\u00a0[latex]3[\/latex], since [latex]y={10}^{x}[\/latex] is increasing. We can examine a graph\u00a0to better estimate the solution.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051924\/CNX_Precalc_Figure_04_03_0022.jpg\" alt=\"Graph of the intersections of the equations y=10^x and y=500.\" width=\"487\" height=\"477\" \/>\r\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 above\u00a0passes the horizontal line test. The exponential function [latex]y={b}^{x}[\/latex] is <strong>one-to-one<\/strong>, so its inverse, [latex]x={b}^{y}[\/latex] is also a function. As is the case with all inverse functions, we simply interchange <em>x<\/em>\u00a0and <em>y<\/em>\u00a0and solve for <em>y<\/em>\u00a0to find the inverse function. To represent <em>y<\/em>\u00a0as a function of <em>x<\/em>, we use a logarithmic function of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. The base <em>b<\/em>\u00a0<strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em>\u00a0to get that number.<\/p>\r\n<p id=\"fs-id1165137404844\">We read a logarithmic expression as, \"The logarithm with base <em>b<\/em>\u00a0of <em>x<\/em>\u00a0is equal to <em>y<\/em>,\" or, simplified, \"log base <em>b<\/em>\u00a0of <em>x<\/em>\u00a0is <em>y<\/em>.\" We can also say, \"<em>b<\/em>\u00a0raised to the power of <em>y<\/em>\u00a0is <em>x<\/em>,\" because logs are exponents. For example, the base\u00a0[latex]2[\/latex] logarithm of\u00a0[latex]32[\/latex] is\u00a0[latex]5[\/latex], because\u00a0[latex]5[\/latex] is the exponent we must apply to\u00a0[latex]2[\/latex] to get\u00a0[latex]32[\/latex]. Since [latex]{2}^{5}=32[\/latex], we can write [latex]{\\mathrm{log}}_{2}32=5[\/latex]. We read this as \"log base\u00a0[latex]2[\/latex] of\u00a0[latex]32[\/latex] is\u00a0[latex]5[\/latex].\"<\/p>\r\n<p id=\"fs-id1165137597501\">We can express the relationship between logarithmic form and its corresponding exponential form as follows:<\/p>\r\n\r\n<div id=\"eip-604\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y\\Leftrightarrow {b}^{y}=x,\\text{}b&gt;0,b\\ne 1[\/latex]<\/div>\r\n<p id=\"fs-id1165137678993\">Note that the base <em>b<\/em>\u00a0is always positive.<span id=\"fs-id1165137696233\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051926\/CNX_Precalc_Figure_04_03_0042.jpg\" alt=\"Think b to the y equals x.\" width=\"487\" height=\"83\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137400957\">Because logarithms are functions, they are 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 <em>x<\/em>.<\/p>\r\n<p id=\"fs-id1165137827516\">We can illustrate the notation of logarithms as follows:<span id=\"fs-id1165137771679\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051928\/CNX_Precalc_Figure_04_03_0032.jpg\" alt=\"logb (c) = a means b to the A power equals C.\" width=\"487\" height=\"101\" \/><\/span><\/p>\r\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>\r\n\r\n<div id=\"fs-id1165137472937\" class=\"note textbox\">\r\n<h3 class=\"title\">Definition of the Logarithmic Function<\/h3>\r\n<p id=\"fs-id1165137704597\">A <strong>logarithm<\/strong> base <em>b<\/em>\u00a0of a positive number <em>x<\/em>\u00a0satisfies the following definition.<\/p>\r\n<p id=\"fs-id1165137584967\">For [latex]x&gt;0,b&gt;0,b\\ne 1[\/latex],<\/p>\r\n\r\n<div id=\"fs-id1165137433829\" class=\"equation\" style=\"text-align: center;\">[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\text{ is the same as }{b}^{y}=x[\/latex]<\/div>\r\n<p id=\"fs-id1165137893373\">where<\/p>\r\n\r\n<ul id=\"fs-id1165135530561\">\r\n \t<li>we read [latex]{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] as, \"the logarithm with base <em>b<\/em>\u00a0of <em>x<\/em>\" or the \"log base <em>b<\/em>\u00a0of <em>x<\/em>.\"<\/li>\r\n \t<li>the logarithm <em>y<\/em>\u00a0is the exponent to which <em>b<\/em>\u00a0must be raised to get <em>x<\/em>.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137547773\">Also, since the logarithmic and exponential functions switch the <em>x<\/em>\u00a0and <em>y<\/em>\u00a0values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,<\/p>\r\n\r\n<ul id=\"fs-id1165137643167\">\r\n \t<li>the domain of the logarithm function with base [latex]b \\text{ is} \\left(0,\\infty \\right)[\/latex].<\/li>\r\n \t<li>the range of the logarithm function with base [latex]b \\text{ is} \\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn our first example, we will convert logarithmic equations into exponential equations.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165137580570\">Write the following logarithmic equations in exponential form.<\/p>\r\n\r\n<ol id=\"fs-id1165137705346\">\r\n \t<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]\r\n[reveal-answer q=\"161275\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161275\"]\r\n<p id=\"fs-id1165137408172\">First, identify the values of <em>b<\/em>,\u00a0<em>y<\/em>, and\u00a0<em>x<\/em>. Then, write the equation in the form [latex]{b}^{y}=x[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1165137705659\">\r\n \t<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]\r\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] can be written as [latex]{6}^{\\frac{1}{2}}=\\sqrt{6}[\/latex].<\/p>\r\n<\/li>\r\n \t<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]\r\n<p id=\"fs-id1165137698078\">Here, <em>b\u00a0<\/em>=[latex]3[\/latex], <em>y\u00a0<\/em>=[latex]2[\/latex], and <em>x\u00a0<\/em>=[latex]9[\/latex]. Therefore, the equation [latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex] can be written as [latex]{3}^{2}=9[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the following video, we present more examples of rewriting logarithmic equations as exponential equations.\r\n\r\nhttps:\/\/youtu.be\/q9_s0wqhIXU\r\n<div id=\"fs-id1165137874700\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137806301\">How To: Given an equation in logarithmic form [latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex], convert it to exponential form<\/h3>\r\n<ol id=\"fs-id1165137641669\">\r\n \t<li>Examine the equation [latex]y={\\mathrm{log}}_{b}x[\/latex] and identify <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>.<\/li>\r\n \t<li>Rewrite [latex]{\\mathrm{log}}_{b}x=y[\/latex] as [latex]{b}^{y}=x[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\n<p id=\"eip-id1549475\">Can we take the logarithm of a negative number? Re-read the definition of a logarithm and formulate an answer. \u00a0Think about the behavior of exponents. \u00a0You can use the textbox below to formulate your ideas before you look at an answer.<\/p>\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"162494\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162494\"]\r\n<p id=\"fs-id1165137653864\">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.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\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, \"To what exponent must\u00a0[latex]2[\/latex]\u00a0be raised in order to get\u00a0[latex]8[\/latex]?\" Because we already know [latex]{2}^{3}=8[\/latex], it follows that [latex]{\\mathrm{log}}_{2}8=3[\/latex].<\/p>\r\n<p id=\"fs-id1165137733822\">Now consider solving [latex]{\\mathrm{log}}_{7}49[\/latex] and [latex]{\\mathrm{log}}_{3}27[\/latex] mentally.<\/p>\r\n\r\n<ul id=\"fs-id1165137937690\">\r\n \t<li>We ask, \"To what exponent must\u00a0[latex]7[\/latex] be raised in order to get\u00a0[latex]49[\/latex]?\" We know [latex]{7}^{2}=49[\/latex]. Therefore, [latex]{\\mathrm{log}}_{7}49=2[\/latex]<\/li>\r\n \t<li>We ask, \"To what exponent must\u00a0[latex]3[\/latex] be raised in order to get\u00a0[latex]27[\/latex]?\" We know [latex]{3}^{3}=27[\/latex]. Therefore, [latex]{\\mathrm{log}}_{3}27=3[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137456358\">Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let us evaluate [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.<\/p>\r\n\r\n<ul id=\"fs-id1165137584208\">\r\n \t<li>We ask, \"To what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}[\/latex]? \" 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>\r\n<\/ul>\r\nIn our first example, we will evaluate logarithms mentally.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.\r\n[reveal-answer q=\"161686\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161686\"]\r\n<p id=\"fs-id1165137611276\">First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[\/latex]. Next, we ask, \"To what exponent must\u00a0[latex]4[\/latex] be raised in order to get\u00a0[latex]64[\/latex]?\"<\/p>\r\nWe know\r\n<div id=\"eip-id1165134583995\" class=\"equation unnumbered\">[latex]{4}^{3}=64[\/latex]<\/div>\r\n<p id=\"fs-id1165137619013\">Therefore,<\/p>\r\n\r\n<div id=\"eip-id1165135606935\" class=\"equation unnumbered\">[latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our first video, we will show more examples of evaluating logarithms mentally; this helps you get familiar with what a logarithm represents.\r\n\r\nhttps:\/\/youtu.be\/dxj5J9OpWGA\r\n\r\nIn our next example, we will evaluate the logarithm of a reciprocal.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex] without using a calculator.\r\n[reveal-answer q=\"534439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"534439\"]\r\n<p id=\"fs-id1165137638179\">First we rewrite the logarithm in exponential form: [latex]{3}^{y}=\\frac{1}{27}[\/latex]. Next, we ask, \"To what exponent must\u00a0[latex]3[\/latex] be raised in order to get [latex]\\frac{1}{27}[\/latex]?\"<\/p>\r\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>\r\n[latex]\\begin{array}{ll}{3}^{-3} &amp; =\\frac{1}{{3}^{3}} \\\\ &amp; =\\frac{1}{27}\\hfill \\end{array}[\/latex]\r\n<p id=\"fs-id1165137585807\">Therefore, [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n\r\n&nbsp;\r\n<h3 id=\"fs-id1165137453770\">How To: Given a logarithm of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], evaluate it mentally<\/h3>\r\n<ol id=\"fs-id1165134079724\">\r\n \t<li>Rewrite the argument <em>x<\/em>\u00a0as a power of <em>b<\/em>: [latex]{b}^{y}=x[\/latex].<\/li>\r\n \t<li>Use previous knowledge of powers of <em>b<\/em>\u00a0to identify <em>y<\/em>\u00a0by asking, \"To what exponent should <em>b<\/em>\u00a0be raised in order to get <em>x<\/em>?\"<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Convert from Exponential to Logarithmic Form<\/h2>\r\nTo convert from exponential form to logarithmic form, we follow the same steps in reverse. We identify the base <em>b<\/em>, exponent <em>x<\/em>, and output <em>y<\/em>. Then we write [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165137804412\">Write the following exponential equations in logarithmic form.<\/p>\r\n\r\n<ol id=\"fs-id1165135192287\">\r\n \t<li>[latex]{2}^{3}=8[\/latex]<\/li>\r\n \t<li>[latex]{5}^{2}=25[\/latex]<\/li>\r\n \t<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"516026\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"516026\"]\r\n<p id=\"fs-id1165137474116\">First, identify the values of <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>. Then, write the equation in the form [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1165137573458\">\r\n \t<li>[latex]{2}^{3}=8[\/latex]\r\n<p id=\"fs-id1165137466396\">Here, [latex]b=2[\/latex], [latex]x=3[\/latex], and\u00a0\u00a0[latex]y=8[\/latex]. Therefore, the equation [latex]{2}^{3}=8[\/latex] can be written as [latex]{\\mathrm{log}}_{2}\\left(8\\right)=3[\/latex].<\/p>\r\n<\/li>\r\n \t<li>[latex]{5}^{2}=25[\/latex]\r\n<p id=\"fs-id1165135193035\">Here,\u00a0\u00a0[latex]b=5[\/latex], [latex]x=2[\/latex], and\u00a0\u00a0[latex]y=25[\/latex]. Therefore, the equation [latex]{5}^{2}=25[\/latex] can be written as [latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex].<\/p>\r\n<\/li>\r\n \t<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]\r\n<p id=\"fs-id1165135187822\">Here, [latex]b=10[\/latex], [latex]x=\u20134[\/latex], and [latex]y=\\frac{1}{10,000}[\/latex]. Therefore, the equation [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex] can be written as [latex]{\\text{log}}_{10}\\left(\\frac{1}{10,000}\\right)=-4[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our last video, we show more examples of writing logarithmic equations as exponential equations.\r\n\r\nhttps:\/\/youtu.be\/9_GPPUWEJQQ\r\n<h2>Summary<\/h2>\r\nThe base <em>b<\/em>\u00a0<strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em>\u00a0to get that number. Logarithmic functions are the inverse of exponential functions, and it is often easier to understand them through this lens.\u00a0We can never take the logarithm of a negative number, therefore\u00a0[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex] is defined for [latex]b&gt;0[\/latex]\r\n\r\n&nbsp;\r\n<p id=\"fs-id1165137748716\">Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.<\/p>\r\n<p id=\"fs-id1165137758495\">Recall that the exponential function is defined as [latex]y={b}^{x}[\/latex] for any real number <em>x<\/em>\u00a0and constant [latex]b&gt;0[\/latex], [latex]b\\ne 1[\/latex], where<\/p>\r\n\r\n<ul id=\"fs-id1165137736024\">\r\n \t<li>The domain of is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n \t<li>The range of is [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135641666\">In the last section we learned that the logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the inverse of the exponential function [latex]y={b}^{x}[\/latex]. So, as inverse functions:<\/p>\r\n\r\n<ul id=\"fs-id1165137656096\">\r\n \t<li>The domain of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the range of [latex]y={b}^{x}[\/latex]: [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n \t<li>The range of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the domain of [latex]y={b}^{x}[\/latex]: [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1165137423048\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135173951\">How To: Given a logarithmic function, identify the domain<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165137823224\">\r\n \t<li>Set up an inequality showing the argument greater than zero.<\/li>\r\n \t<li>Solve for <em>x<\/em>.<\/li>\r\n \t<li>Write the domain in interval notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn our first example, we will show how to identify the domain of a logarithmic function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhat is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex]?\r\n[reveal-answer q=\"370398\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"370398\"]\r\n<p id=\"fs-id1165137693442\">The logarithmic function is defined only when the input is positive, so this function is defined when [latex]x+3&gt;0[\/latex]. Solving this inequality,<\/p>\r\n\r\n<div id=\"eip-id1165135381135\" class=\"equation unnumbered\">[latex]\\begin{array}{c}x+3&gt;0\\hfill &amp; \\text{The input must be positive}.\\hfill \\\\ x&gt;-3\\hfill &amp; \\text{Subtract 3}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137638183\">The domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex] is [latex]\\left(-3,\\infty \\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere is another example of how to identify the domain of a logarithmic function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhat is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex]?\r\n[reveal-answer q=\"275313\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"275313\"]\r\n<p id=\"fs-id1165137780875\">The logarithmic function is defined only when the input is positive, so this function is defined when [latex]5 - 2x&gt;0[\/latex]. Solving this inequality,<\/p>\r\n\r\n<div id=\"eip-id1165135470032\" class=\"equation unnumbered\">[latex]\\begin{array}{c}5 - 2x&gt;0\\hfill &amp; \\text{The input must be positive}.\\hfill \\\\ -2x&gt;-5\\hfill &amp; \\text{Subtract }5.\\hfill \\\\ x&lt;\\frac{5}{2}\\hfill &amp; \\text{Divide by }-2\\text{ and switch the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137656879\">The domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex] is [latex]\\left(-\\infty ,\\frac{5}{2}\\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>\u00a0Graph Logarithmic Functions<\/h2>\r\n<p id=\"fs-id1165135194555\">Creating a graphical representation of most functions\u00a0gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the <em>cause<\/em> for an <em>effect<\/em>.<\/p>\r\n<p id=\"fs-id1165137603580\">To illustrate, suppose we invest $2500 in an account that offers an annual interest rate of\u00a0[latex]5\\%[\/latex], compounded continuously. We already know that the balance in our account for any year <em>t<\/em>\u00a0can be found with the equation [latex]A=2500{e}^{0.05t}[\/latex].<\/p>\r\nBut what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? The image below shows this point on the logarithmic graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051930\/CNX_Precalc_Figure_04_04_0012.jpg\" alt=\"A graph titled,\" width=\"900\" height=\"459\" \/> A logarithmic graph showing how many years it will take for an account balance to double.[\/caption]\r\n<p id=\"fs-id1165134104063\">Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] along with all its transformations: shifts, stretches, compressions, and reflections.<\/p>\r\n<p id=\"fs-id1165137679088\">We begin with the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Because every logarithmic function of this form is the inverse of an exponential function of the form [latex]y={b}^{x}[\/latex], their graphs will be reflections of each other across the line [latex]y=x[\/latex]. To illustrate this, we can observe the relationship between the input and output values of [latex]y={2}^{x}[\/latex] and its equivalent [latex]x={\\mathrm{log}}_{2}\\left(y\\right)[\/latex] in the table below.<\/p>\r\n\r\n<table summary=\"Three rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>[latex]\u20133[\/latex]<\/td>\r\n<td>[latex]\u20132[\/latex]<\/td>\r\n<td>[latex]\u20131[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\r\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\r\n<td>[latex]\u20133[\/latex]<\/td>\r\n<td>[latex]\u20132[\/latex]<\/td>\r\n<td>[latex]\u20131[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135509175\">Using the inputs and outputs from the table above, we can build another table to observe the relationship between points on the graphs of the inverse functions [latex]f\\left(x\\right)={2}^{x}[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/p>\r\n\r\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(0,1\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(1,2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(3,8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td>[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(1,0\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(2,1\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(4,2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(8,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137761335\">As we would expect, the <em>x<\/em>- and <em>y<\/em>-coordinates are reversed for the inverse functions. The figure below\u00a0shows the graph of <em>f<\/em>\u00a0and <em>g<\/em>.<\/p>\r\n\r\n<figure class=\"small\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051932\/CNX_Precalc_Figure_04_04_0022.jpg\" alt=\"Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.\" \/><\/figure>\r\nNotice that the graphs of [latex]f\\left(x\\right)={2}^{x}[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] are reflections about the line <em>y\u00a0<\/em>= <em>x<\/em>.\r\n<p id=\"fs-id1165137406913\">Observe the following from the graph:<\/p>\r\n\r\n<ul id=\"fs-id1165137408405\">\r\n \t<li>[latex]f\\left(x\\right)={2}^{x}[\/latex] has a <em>y<\/em>-intercept at [latex]\\left(0,1\\right)[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] has an <em>x<\/em>-intercept at [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t<li>The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex], [latex]\\left(-\\infty ,\\infty \\right)[\/latex], is the same as the range of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/li>\r\n \t<li>The range of [latex]f\\left(x\\right)={2}^{x}[\/latex], [latex]\\left(0,\\infty \\right)[\/latex], is the same as the domain of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/li>\r\n<\/ul>\r\n<h3 class=\"title\">A General Note: Characteristics of the Graph of the Parent Function, <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\r\n<p id=\"fs-id1165135520250\">For any real number <em>x<\/em>\u00a0and constant <em>b\u00a0<\/em>&gt; 0, [latex]b\\ne 1[\/latex], we can see the following characteristics in the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]:<\/p>\r\n\r\n<ul id=\"fs-id1165137400150\">\r\n \t<li>one-to-one function<\/li>\r\n \t<li>vertical asymptote: <em>x\u00a0<\/em>=[latex]0[\/latex]<\/li>\r\n \t<li>domain: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\r\n \t<li>range: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/li>\r\n \t<li><em>x-<\/em>intercept: [latex]\\left(1,0\\right)[\/latex] and key point [latex]\\left(b,1\\right)[\/latex]<\/li>\r\n \t<li><em>y<\/em>-intercept: none<\/li>\r\n \t<li>increasing if [latex]b&gt;1[\/latex]<\/li>\r\n \t<li>decreasing if [latex]0 \\lt b \\lt 1[\/latex]<\/li>\r\n<\/ul>\r\n<figure id=\"CNX_Precalc_Figure_04_04_003\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051934\/CNX_Precalc_Figure_04_04_003G2.jpg\" alt=\"&quot;Two\" \/><\/figure>\r\nThe figure below shows how changing the base <em>b<\/em>\u00a0in [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (<em>Note:<\/em> recall that the function [latex]\\mathrm{ln}\\left(x\\right)[\/latex] has base [latex]e\\approx \\text{2}.\\text{718.)}[\/latex]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051936\/CNX_Precalc_Figure_04_04_0042.jpg\" alt=\"Graph of three equations: y=log_2(x) in blue, y=ln(x) in orange, and y=log(x) in red. The y-axis is the asymptote.\" width=\"487\" height=\"363\" \/> The graphs of three logarithmic functions with different bases, all greater than 1.[\/caption]\r\n\r\nIn our next example, we will graph a logarithmic function of the form\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGraph [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]. State the domain and range.\r\n[reveal-answer q=\"486007\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"486007\"]\r\n<p id=\"fs-id1165137501970\">Before graphing, identify the behavior and key points for the graph.<\/p>\r\n\r\n<ul id=\"fs-id1165135497154\">\r\n \t<li>Since [latex]b=5[\/latex] is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical line\u00a0[latex]x=0[\/latex], and the right tail will increase slowly without bound.<\/li>\r\n \t<li>The <em>x<\/em>-intercept is [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t<li>The key point [latex]\\left(5,1\\right)[\/latex] is on the graph.<\/li>\r\n \t<li>We plot and label the points, and draw a smooth curve through the points.<\/li>\r\n \t<li>The domain is [latex]\\left(0,\\infty \\right)[\/latex], and the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<figure id=\"CNX_Precalc_Figure_04_04_005\" class=\"small\"><span id=\"fs-id1165135508394\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051938\/CNX_Precalc_Figure_04_04_0052.jpg\" alt=\"Graph of f(x)=log_5(x) with labeled points at (1, 0) and (5, 1). The y-axis is the asymptote.\" \/><\/span><\/figure>\r\n<p id=\"fs-id1165135697920\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3 id=\"fs-id1165137805513\">How To: Given a logarithmic function of the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph the function<\/h3>\r\n<ol id=\"fs-id1165135435529\">\r\n \t<li>Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t<li>Plot the key point [latex]\\left(b,1\\right)[\/latex].<\/li>\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n \t<li>State the domain, [latex]\\left(0,\\infty \\right)[\/latex], and the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nTo define the domain of a logarithmic function algebraically, set the argument greater than zero and solve. To plot a logarithmic function, it is easiest to find and plot the x-intercept and the key point [latex]\\left(b,1\\right)[\/latex].","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate logarithmic expressions using the definition of logarithms.<\/li>\n<li>Graph logarithmic functions by creating a table of values.<\/li>\n<li>Find the equation of the asymptote for a logarithmic function.<\/li>\n<li>Find the domain and range of a logarithmic function using its graph and write them in interval notation.<\/li>\n<li>Rewrite exponential equations in logarithmic form.<\/li>\n<li>Rewrite logarithmic equations in exponential form.<\/li>\n<\/ul>\n<\/div>\n<figure id=\"CNX_Precalc_Figure_04_03_001\" class=\"small\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051922\/CNX_Precalc_Figure_04_03_0012.jpg\" alt=\"Photo of the aftermath of the earthquake in Japan with a focus on the Japanese flag.\" width=\"488\" height=\"325\" \/><\/p>\n<p class=\"wp-caption-text\">Devastation of the earthquake which happened on March 11, 2011 in Honshu, Japan. (credit: Daniel Pierce)<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1165137557013\">In [latex]2010[\/latex], a major earthquake struck Haiti, destroying or damaging over\u00a0[latex]285,000[\/latex] homes.<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary. Accessed\u00a0[latex]3\/4\/2013[\/latex].\" id=\"return-footnote-221-1\" href=\"#footnote-221-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over\u00a0[latex]332,000[\/latex] buildings,<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary. Accessed\u00a0[latex]3\/4\/2013[\/latex].\" id=\"return-footnote-221-2\" href=\"#footnote-221-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>\u00a0like those shown in the picture above. Even though both caused substantial damage, the earthquake in\u00a0[latex]2011[\/latex] was\u00a0[latex]100[\/latex] 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\u00a0[latex]7.0[\/latex] on the Richter Scale<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/. Accessed\u00a0[latex]3\/4\/2013[\/latex].\" id=\"return-footnote-221-3\" href=\"#footnote-221-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>\u00a0whereas the Japanese earthquake registered a\u00a0[latex]9.0[\/latex].<a class=\"footnote\" title=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#details. Accessed\u00a0[latex]3\/4\/2013[\/latex].\" id=\"return-footnote-221-4\" href=\"#footnote-221-4\" 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\u00a0[latex]4[\/latex]. 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<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\u00a0[latex]500[\/latex] 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 <strong>Richter Scale<\/strong>. How would we solve for [latex]x[\/latex]?<\/p>\n<p>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]\u00a0must be some value between\u00a0[latex]2[\/latex] and\u00a0[latex]3[\/latex], since [latex]y={10}^{x}[\/latex] is increasing. We can examine a graph\u00a0to better estimate the solution.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051924\/CNX_Precalc_Figure_04_03_0022.jpg\" alt=\"Graph of the intersections of the equations y=10^x and y=500.\" width=\"487\" height=\"477\" \/><\/p>\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 above\u00a0passes the horizontal line test. The exponential function [latex]y={b}^{x}[\/latex] is <strong>one-to-one<\/strong>, so its inverse, [latex]x={b}^{y}[\/latex] is also a function. As is the case with all inverse functions, we simply interchange <em>x<\/em>\u00a0and <em>y<\/em>\u00a0and solve for <em>y<\/em>\u00a0to find the inverse function. To represent <em>y<\/em>\u00a0as a function of <em>x<\/em>, we use a logarithmic function of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. The base <em>b<\/em>\u00a0<strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em>\u00a0to get that number.<\/p>\n<p id=\"fs-id1165137404844\">We read a logarithmic expression as, &#8220;The logarithm with base <em>b<\/em>\u00a0of <em>x<\/em>\u00a0is equal to <em>y<\/em>,&#8221; or, simplified, &#8220;log base <em>b<\/em>\u00a0of <em>x<\/em>\u00a0is <em>y<\/em>.&#8221; We can also say, &#8220;<em>b<\/em>\u00a0raised to the power of <em>y<\/em>\u00a0is <em>x<\/em>,&#8221; because logs are exponents. For example, the base\u00a0[latex]2[\/latex] logarithm of\u00a0[latex]32[\/latex] is\u00a0[latex]5[\/latex], because\u00a0[latex]5[\/latex] is the exponent we must apply to\u00a0[latex]2[\/latex] to get\u00a0[latex]32[\/latex]. Since [latex]{2}^{5}=32[\/latex], we can write [latex]{\\mathrm{log}}_{2}32=5[\/latex]. We read this as &#8220;log base\u00a0[latex]2[\/latex] of\u00a0[latex]32[\/latex] is\u00a0[latex]5[\/latex].&#8221;<\/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=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y\\Leftrightarrow {b}^{y}=x,\\text{}b>0,b\\ne 1[\/latex]<\/div>\n<p id=\"fs-id1165137678993\">Note that the base <em>b<\/em>\u00a0is always positive.<span id=\"fs-id1165137696233\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051926\/CNX_Precalc_Figure_04_03_0042.jpg\" alt=\"Think b to the y equals x.\" width=\"487\" height=\"83\" \/><\/span><\/p>\n<p id=\"fs-id1165137400957\">Because logarithms are functions, they are 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 <em>x<\/em>.<\/p>\n<p id=\"fs-id1165137827516\">We can illustrate the notation of logarithms as follows:<span id=\"fs-id1165137771679\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051928\/CNX_Precalc_Figure_04_03_0032.jpg\" alt=\"logb (c) = a means b to the A power equals C.\" width=\"487\" height=\"101\" \/><\/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=\"note textbox\">\n<h3 class=\"title\">Definition of the Logarithmic Function<\/h3>\n<p id=\"fs-id1165137704597\">A <strong>logarithm<\/strong> base <em>b<\/em>\u00a0of a positive number <em>x<\/em>\u00a0satisfies 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\" class=\"equation\" style=\"text-align: center;\">[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\text{ is the same as }{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, &#8220;the logarithm with base <em>b<\/em>\u00a0of <em>x<\/em>&#8221; or the &#8220;log base <em>b<\/em>\u00a0of <em>x<\/em>.&#8221;<\/li>\n<li>the logarithm <em>y<\/em>\u00a0is the exponent to which <em>b<\/em>\u00a0must be raised to get <em>x<\/em>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137547773\">Also, since the logarithmic and exponential functions switch the <em>x<\/em>\u00a0and <em>y<\/em>\u00a0values, 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<p>In our first example, we will convert logarithmic equations into exponential equations.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165137580570\">Write the following logarithmic equations in exponential form.<\/p>\n<ol id=\"fs-id1165137705346\">\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]\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q161275\">Show Solution<\/span><\/p>\n<div id=\"q161275\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137408172\">First, identify the values of <em>b<\/em>,\u00a0<em>y<\/em>, and\u00a0<em>x<\/em>. Then, write the equation in the form [latex]{b}^{y}=x[\/latex].<\/p>\n<ol id=\"fs-id1165137705659\">\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] can be written as [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, <em>b\u00a0<\/em>=[latex]3[\/latex], <em>y\u00a0<\/em>=[latex]2[\/latex], and <em>x\u00a0<\/em>=[latex]9[\/latex]. Therefore, the equation [latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex] can be written as [latex]{3}^{2}=9[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<p>In the following video, we present more examples of rewriting logarithmic equations as exponential equations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Write Logarithmic Equations as Exponential Equations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q9_s0wqhIXU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"fs-id1165137874700\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137806301\">How To: Given an equation in logarithmic form [latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex], convert it to exponential form<\/h3>\n<ol id=\"fs-id1165137641669\">\n<li>Examine the equation [latex]y={\\mathrm{log}}_{b}x[\/latex] and identify <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>.<\/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 key-takeaways\">\n<h3>Think About It<\/h3>\n<p id=\"eip-id1549475\">Can we take the logarithm of a negative number? Re-read the definition of a logarithm and formulate an answer. \u00a0Think about the behavior of exponents. \u00a0You can use the textbox below to formulate your ideas before you look at an answer.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162494\">Show Solution<\/span><\/p>\n<div id=\"q162494\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137653864\">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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\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, &#8220;To what exponent must\u00a0[latex]2[\/latex]\u00a0be raised in order to get\u00a0[latex]8[\/latex]?&#8221; 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, &#8220;To what exponent must\u00a0[latex]7[\/latex] be raised in order to get\u00a0[latex]49[\/latex]?&#8221; We know [latex]{7}^{2}=49[\/latex]. Therefore, [latex]{\\mathrm{log}}_{7}49=2[\/latex]<\/li>\n<li>We ask, &#8220;To what exponent must\u00a0[latex]3[\/latex] be raised in order to get\u00a0[latex]27[\/latex]?&#8221; 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 us evaluate [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.<\/p>\n<ul id=\"fs-id1165137584208\">\n<li>We ask, &#8220;To what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}[\/latex]? &#8221; 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<p>In our first example, we will evaluate logarithms mentally.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q161686\">Show Solution<\/span><\/p>\n<div id=\"q161686\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137611276\">First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[\/latex]. Next, we ask, &#8220;To what exponent must\u00a0[latex]4[\/latex] be raised in order to get\u00a0[latex]64[\/latex]?&#8221;<\/p>\n<p>We know<\/p>\n<div id=\"eip-id1165134583995\" class=\"equation unnumbered\">[latex]{4}^{3}=64[\/latex]<\/div>\n<p id=\"fs-id1165137619013\">Therefore,<\/p>\n<div id=\"eip-id1165135606935\" class=\"equation unnumbered\">[latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In our first video, we will show more examples of evaluating logarithms mentally; this helps you get familiar with what a logarithm represents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Evaluate Logarithms Without a Calculator - Whole Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/dxj5J9OpWGA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In our next example, we will evaluate the logarithm of a reciprocal.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q534439\">Show Solution<\/span><\/p>\n<div id=\"q534439\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137638179\">First we rewrite the logarithm in exponential form: [latex]{3}^{y}=\\frac{1}{27}[\/latex]. Next, we ask, &#8220;To what exponent must\u00a0[latex]3[\/latex] be raised in order to get [latex]\\frac{1}{27}[\/latex]?&#8221;<\/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<p>[latex]\\begin{array}{ll}{3}^{-3} & =\\frac{1}{{3}^{3}} \\\\ & =\\frac{1}{27}\\hfill \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137585807\">Therefore, [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<p>&nbsp;<\/p>\n<h3 id=\"fs-id1165137453770\">How To: Given a logarithm of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], evaluate it mentally<\/h3>\n<ol id=\"fs-id1165134079724\">\n<li>Rewrite the argument <em>x<\/em>\u00a0as a power of <em>b<\/em>: [latex]{b}^{y}=x[\/latex].<\/li>\n<li>Use previous knowledge of powers of <em>b<\/em>\u00a0to identify <em>y<\/em>\u00a0by asking, &#8220;To what exponent should <em>b<\/em>\u00a0be raised in order to get <em>x<\/em>?&#8221;<\/li>\n<\/ol>\n<\/div>\n<h2>Convert from Exponential to Logarithmic Form<\/h2>\n<p>To convert from exponential form to logarithmic form, we follow the same steps in reverse. We identify the base <em>b<\/em>, exponent <em>x<\/em>, and output <em>y<\/em>. Then we write [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165137804412\">Write the following exponential equations in logarithmic form.<\/p>\n<ol id=\"fs-id1165135192287\">\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q516026\">Show Solution<\/span><\/p>\n<div id=\"q516026\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137474116\">First, identify the values of <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>. Then, write the equation in the form [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\n<ol id=\"fs-id1165137573458\">\n<li>[latex]{2}^{3}=8[\/latex]\n<p id=\"fs-id1165137466396\">Here, [latex]b=2[\/latex], [latex]x=3[\/latex], and\u00a0\u00a0[latex]y=8[\/latex]. Therefore, the equation [latex]{2}^{3}=8[\/latex] can be written as [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,\u00a0\u00a0[latex]b=5[\/latex], [latex]x=2[\/latex], and\u00a0\u00a0[latex]y=25[\/latex]. Therefore, the equation [latex]{5}^{2}=25[\/latex] can be written as [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=\u20134[\/latex], and [latex]y=\\frac{1}{10,000}[\/latex]. Therefore, the equation [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex] can be written as [latex]{\\text{log}}_{10}\\left(\\frac{1}{10,000}\\right)=-4[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In our last video, we show more examples of writing logarithmic equations as exponential equations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Write Exponential Equations as Logarithmic Equations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9_GPPUWEJQQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>The base <em>b<\/em>\u00a0<strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em>\u00a0to get that number. Logarithmic functions are the inverse of exponential functions, and it is often easier to understand them through this lens.\u00a0We can never take the logarithm of a negative number, therefore\u00a0[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex] is defined for [latex]b>0[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137748716\">Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.<\/p>\n<p id=\"fs-id1165137758495\">Recall that the exponential function is defined as [latex]y={b}^{x}[\/latex] for any real number <em>x<\/em>\u00a0and constant [latex]b>0[\/latex], [latex]b\\ne 1[\/latex], where<\/p>\n<ul id=\"fs-id1165137736024\">\n<li>The domain of is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<li>The range of is [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<p id=\"fs-id1165135641666\">In the last section we learned that the logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the inverse of the exponential function [latex]y={b}^{x}[\/latex]. So, as inverse functions:<\/p>\n<ul id=\"fs-id1165137656096\">\n<li>The domain of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the range of [latex]y={b}^{x}[\/latex]: [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n<li>The range of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the domain of [latex]y={b}^{x}[\/latex]: [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<div id=\"fs-id1165137423048\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135173951\">How To: Given a logarithmic function, identify the domain<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137823224\">\n<li>Set up an inequality showing the argument greater than zero.<\/li>\n<li>Solve for <em>x<\/em>.<\/li>\n<li>Write the domain in interval notation.<\/li>\n<\/ol>\n<\/div>\n<p>In our first example, we will show how to identify the domain of a logarithmic function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q370398\">Show Solution<\/span><\/p>\n<div id=\"q370398\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137693442\">The logarithmic function is defined only when the input is positive, so this function is defined when [latex]x+3>0[\/latex]. Solving this inequality,<\/p>\n<div id=\"eip-id1165135381135\" class=\"equation unnumbered\">[latex]\\begin{array}{c}x+3>0\\hfill & \\text{The input must be positive}.\\hfill \\\\ x>-3\\hfill & \\text{Subtract 3}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137638183\">The domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex] is [latex]\\left(-3,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here is another example of how to identify the domain of a logarithmic function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q275313\">Show Solution<\/span><\/p>\n<div id=\"q275313\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137780875\">The logarithmic function is defined only when the input is positive, so this function is defined when [latex]5 - 2x>0[\/latex]. Solving this inequality,<\/p>\n<div id=\"eip-id1165135470032\" class=\"equation unnumbered\">[latex]\\begin{array}{c}5 - 2x>0\\hfill & \\text{The input must be positive}.\\hfill \\\\ -2x>-5\\hfill & \\text{Subtract }5.\\hfill \\\\ x<\\frac{5}{2}\\hfill & \\text{Divide by }-2\\text{ and switch the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137656879\">The domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex] is [latex]\\left(-\\infty ,\\frac{5}{2}\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Graph Logarithmic Functions<\/h2>\n<p id=\"fs-id1165135194555\">Creating a graphical representation of most functions\u00a0gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the <em>cause<\/em> for an <em>effect<\/em>.<\/p>\n<p id=\"fs-id1165137603580\">To illustrate, suppose we invest $2500 in an account that offers an annual interest rate of\u00a0[latex]5\\%[\/latex], compounded continuously. We already know that the balance in our account for any year <em>t<\/em>\u00a0can be found with the equation [latex]A=2500{e}^{0.05t}[\/latex].<\/p>\n<p>But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? The image below shows this point on the logarithmic graph.<\/p>\n<div style=\"width: 910px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051930\/CNX_Precalc_Figure_04_04_0012.jpg\" alt=\"A graph titled,\" width=\"900\" height=\"459\" \/><\/p>\n<p class=\"wp-caption-text\">A logarithmic graph showing how many years it will take for an account balance to double.<\/p>\n<\/div>\n<p id=\"fs-id1165134104063\">Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] along with all its transformations: shifts, stretches, compressions, and reflections.<\/p>\n<p id=\"fs-id1165137679088\">We begin with the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Because every logarithmic function of this form is the inverse of an exponential function of the form [latex]y={b}^{x}[\/latex], their graphs will be reflections of each other across the line [latex]y=x[\/latex]. To illustrate this, we can observe the relationship between the input and output values of [latex]y={2}^{x}[\/latex] and its equivalent [latex]x={\\mathrm{log}}_{2}\\left(y\\right)[\/latex] in the table below.<\/p>\n<table summary=\"Three rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>[latex]\u20133[\/latex]<\/td>\n<td>[latex]\u20132[\/latex]<\/td>\n<td>[latex]\u20131[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\n<td>[latex]\u20133[\/latex]<\/td>\n<td>[latex]\u20132[\/latex]<\/td>\n<td>[latex]\u20131[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135509175\">Using the inputs and outputs from the table above, we can build another table to observe the relationship between points on the graphs of the inverse functions [latex]f\\left(x\\right)={2}^{x}[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/p>\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\n<td>[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\n<td>[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\n<td>[latex]\\left(0,1\\right)[\/latex]<\/td>\n<td>[latex]\\left(1,2\\right)[\/latex]<\/td>\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td>[latex]\\left(3,8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\n<td>[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\n<td>[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\n<td>[latex]\\left(1,0\\right)[\/latex]<\/td>\n<td>[latex]\\left(2,1\\right)[\/latex]<\/td>\n<td>[latex]\\left(4,2\\right)[\/latex]<\/td>\n<td>[latex]\\left(8,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137761335\">As we would expect, the <em>x<\/em>&#8211; and <em>y<\/em>-coordinates are reversed for the inverse functions. The figure below\u00a0shows the graph of <em>f<\/em>\u00a0and <em>g<\/em>.<\/p>\n<figure class=\"small\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051932\/CNX_Precalc_Figure_04_04_0022.jpg\" alt=\"Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.\" \/><\/figure>\n<p>Notice that the graphs of [latex]f\\left(x\\right)={2}^{x}[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] are reflections about the line <em>y\u00a0<\/em>= <em>x<\/em>.<\/p>\n<p id=\"fs-id1165137406913\">Observe the following from the graph:<\/p>\n<ul id=\"fs-id1165137408405\">\n<li>[latex]f\\left(x\\right)={2}^{x}[\/latex] has a <em>y<\/em>-intercept at [latex]\\left(0,1\\right)[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] has an <em>x<\/em>-intercept at [latex]\\left(1,0\\right)[\/latex].<\/li>\n<li>The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex], [latex]\\left(-\\infty ,\\infty \\right)[\/latex], is the same as the range of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/li>\n<li>The range of [latex]f\\left(x\\right)={2}^{x}[\/latex], [latex]\\left(0,\\infty \\right)[\/latex], is the same as the domain of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/li>\n<\/ul>\n<h3 class=\"title\">A General Note: Characteristics of the Graph of the Parent Function, <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p id=\"fs-id1165135520250\">For any real number <em>x<\/em>\u00a0and constant <em>b\u00a0<\/em>&gt; 0, [latex]b\\ne 1[\/latex], we can see the following characteristics in the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]:<\/p>\n<ul id=\"fs-id1165137400150\">\n<li>one-to-one function<\/li>\n<li>vertical asymptote: <em>x\u00a0<\/em>=[latex]0[\/latex]<\/li>\n<li>domain: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\n<li>range: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/li>\n<li><em>x-<\/em>intercept: [latex]\\left(1,0\\right)[\/latex] and key point [latex]\\left(b,1\\right)[\/latex]<\/li>\n<li><em>y<\/em>-intercept: none<\/li>\n<li>increasing if [latex]b>1[\/latex]<\/li>\n<li>decreasing if [latex]0 \\lt b \\lt 1[\/latex]<\/li>\n<\/ul>\n<figure id=\"CNX_Precalc_Figure_04_04_003\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051934\/CNX_Precalc_Figure_04_04_003G2.jpg\" alt=\"&quot;Two\" \/><\/figure>\n<p>The figure below shows how changing the base <em>b<\/em>\u00a0in [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (<em>Note:<\/em> recall that the function [latex]\\mathrm{ln}\\left(x\\right)[\/latex] has base [latex]e\\approx \\text{2}.\\text{718.)}[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051936\/CNX_Precalc_Figure_04_04_0042.jpg\" alt=\"Graph of three equations: y=log_2(x) in blue, y=ln(x) in orange, and y=log(x) in red. The y-axis is the asymptote.\" width=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\">The graphs of three logarithmic functions with different bases, all greater than 1.<\/p>\n<\/div>\n<p>In our next example, we will graph a logarithmic function of the form\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]. State the domain and range.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q486007\">Show Solution<\/span><\/p>\n<div id=\"q486007\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137501970\">Before graphing, identify the behavior and key points for the graph.<\/p>\n<ul id=\"fs-id1165135497154\">\n<li>Since [latex]b=5[\/latex] is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical line\u00a0[latex]x=0[\/latex], and the right tail will increase slowly without bound.<\/li>\n<li>The <em>x<\/em>-intercept is [latex]\\left(1,0\\right)[\/latex].<\/li>\n<li>The key point [latex]\\left(5,1\\right)[\/latex] is on the graph.<\/li>\n<li>We plot and label the points, and draw a smooth curve through the points.<\/li>\n<li>The domain is [latex]\\left(0,\\infty \\right)[\/latex], and the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<figure id=\"CNX_Precalc_Figure_04_04_005\" class=\"small\"><span id=\"fs-id1165135508394\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051938\/CNX_Precalc_Figure_04_04_0052.jpg\" alt=\"Graph of f(x)=log_5(x) with labeled points at (1, 0) and (5, 1). The y-axis is the asymptote.\" \/><\/span><\/figure>\n<p id=\"fs-id1165135697920\"><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3 id=\"fs-id1165137805513\">How To: Given a logarithmic function of the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph the function<\/h3>\n<ol id=\"fs-id1165135435529\">\n<li>Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/li>\n<li>Plot the key point [latex]\\left(b,1\\right)[\/latex].<\/li>\n<li>Draw a smooth curve through the points.<\/li>\n<li>State the domain, [latex]\\left(0,\\infty \\right)[\/latex], and the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<h2>Summary<\/h2>\n<p>To define the domain of a logarithmic function algebraically, set the argument greater than zero and solve. To plot a logarithmic function, it is easiest to find and plot the x-intercept and the key point [latex]\\left(b,1\\right)[\/latex].<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-221\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Ex: Write Exponential Equations as Logarithmic Equations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9_GPPUWEJQQ\">https:\/\/youtu.be\/9_GPPUWEJQQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><li>Ex: Write Logarithmic Equations as Exponential Equations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q9_s0wqhIXU\">https:\/\/youtu.be\/q9_s0wqhIXU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-221-1\"><a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary\" target=\"_blank\" rel=\"noopener\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/#summary<\/a>. Accessed\u00a0[latex]3\/4\/2013[\/latex]. <a href=\"#return-footnote-221-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-221-2\"><a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary\" target=\"_blank\" rel=\"noopener\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#summary<\/a>. Accessed\u00a0[latex]3\/4\/2013[\/latex]. <a href=\"#return-footnote-221-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-221-3\"><a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/\" target=\"_blank\" rel=\"noopener\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2010\/us2010rja6\/<\/a>. Accessed\u00a0[latex]3\/4\/2013[\/latex]. <a href=\"#return-footnote-221-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-221-4\"><a href=\"http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#details\" target=\"_blank\" rel=\"noopener\">http:\/\/earthquake.usgs.gov\/earthquakes\/eqinthenews\/2011\/usc0001xgp\/#details<\/a>. Accessed\u00a0[latex]3\/4\/2013[\/latex]. <a href=\"#return-footnote-221-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":395986,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"cc\",\"description\":\"Ex: Write Logarithmic Equations as Exponential Equations\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/q9_s0wqhIXU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex: Write Exponential Equations as Logarithmic Equations\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/9_GPPUWEJQQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"17bf92f8-a510-4ed7-a14b-cecaaddd1b58","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-221","chapter","type-chapter","status-publish","hentry"],"part":225,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/221","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/221\/revisions"}],"predecessor-version":[{"id":1659,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/221\/revisions\/1659"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/parts\/225"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/221\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/media?parent=221"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=221"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/contributor?post=221"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/license?post=221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}