{"id":15992,"date":"2019-09-26T16:05:43","date_gmt":"2019-09-26T16:05:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/introduction-to-graphs-of-logarithmic-functions\/"},"modified":"2024-05-02T16:54:06","modified_gmt":"2024-05-02T16:54:06","slug":"introduction-to-graphs-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/introduction-to-graphs-of-logarithmic-functions\/","title":{"raw":"Graphing Logarithmic Functions","rendered":"Graphing Logarithmic Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>LEARNING Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the domain of a logarithmic function<\/li>\r\n \t<li>Graph logarithmic functions<\/li>\r\n<\/ul>\r\n<\/div>\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\nTherefore, when finding the domain of a logarithmic function, it is important to remember that the domain consists\u00a0<em>only of positive real numbers<\/em>. That is, the value you are applying the logarithmic function to, also known as the argument of the logarithmic function, must be greater than zero.\r\n\r\nFor example, consider [latex]y={\\mathrm{log}}_{4}\\left(2x-3\\right)[\/latex].\u00a0 This function is defined for any values of x such that the argument, in this case [latex]2x-3[\/latex], is greater than zero.\u00a0 To find the domain, we set up an inequality and solve for x:\r\n\r\n[latex]\\begin{array}{c}2x-3&gt;0\\hfill &amp; \\text{The argument must be positive}.\\hfill \\\\ 2x&gt;3\\hfill &amp; \\text{Add 3}.\\hfill \\\\ x&gt;1.5\\hfill &amp; \\text{Divide by 2}.\\hfill \\end{array}[\/latex]\r\n\r\nIn interval notation, the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x-3\\right)[\/latex] is [latex]\\left(1.5,\\infty \\right)[\/latex].\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 argument 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\" style=\"text-align: center;\">[latex]\\begin{array}{c}x+3&gt;0\\hfill &amp; \\text{The argument 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 argument 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\" style=\"text-align: center;\">[latex]\\begin{array}{c}5 - 2x&gt;0\\hfill &amp; \\text{The argument 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 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, Logarithmic Model Showing Years as a Function of the Balance in the Account. The horizontal axis runs from 0 to 6000 and is labeled account balance. The vertical axis runs from 0 to 20 and is labeled years. The graph has an x-intercept at (2500, 0) and grows logarithmically. An arrow labels a point on the graph, at which the balance reaches 5000 dollars near year 14.\" 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 style=\"width: 70%;\" summary=\"Three rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 30.3725%;\"><em><strong>x<\/strong><\/em><\/td>\r\n<td style=\"width: 15.6275%; text-align: center;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 11%; text-align: center;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 11%; text-align: center;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 6%; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 10%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 6%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 1%; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 30.3725%;\"><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\r\n<td style=\"width: 15.6275%; text-align: center;\">[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 11%; text-align: center;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 11%; text-align: center;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 6%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 10%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 6%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 1%; text-align: center;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 30.3725%;\"><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 15.6275%; text-align: center;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 11%; text-align: center;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 11%; text-align: center;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 6%; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 10%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 6%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 1%; text-align: center;\">[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 style=\"width: 70%;\" 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>.\u00a0 This is a special property of the relationship between any function and its inverse.\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 class=\"alignnone\" 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=\"Two side-by-side logarithmic graphs. The left graph is of the general form equation f of x equals log base b of x where b is greater than 1. The graph has a vertical asymptote of x equals 0 and points of (1, 0) and (b, 1). f of x approaches negative infinity as x approaches 0. The right graph is of the general form equation f of x equals log base b of x where b is between 0 and 1. The graph has a vertical asymptote of x equals 0 and points of (b, 1) and (1, 0). f of x approaches infinity as x approaches 0.\" width=\"824\" height=\"367\" \/><\/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\" style=\"text-align: left;\">[\/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=\"bcc-box bcc-highlight\">\n<h3>LEARNING Outcomes<\/h3>\n<ul>\n<li>Identify the domain of a logarithmic function<\/li>\n<li>Graph logarithmic functions<\/li>\n<\/ul>\n<\/div>\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<p>Therefore, when finding the domain of a logarithmic function, it is important to remember that the domain consists\u00a0<em>only of positive real numbers<\/em>. That is, the value you are applying the logarithmic function to, also known as the argument of the logarithmic function, must be greater than zero.<\/p>\n<p>For example, consider [latex]y={\\mathrm{log}}_{4}\\left(2x-3\\right)[\/latex].\u00a0 This function is defined for any values of x such that the argument, in this case [latex]2x-3[\/latex], is greater than zero.\u00a0 To find the domain, we set up an inequality and solve for x:<\/p>\n<p>[latex]\\begin{array}{c}2x-3>0\\hfill & \\text{The argument must be positive}.\\hfill \\\\ 2x>3\\hfill & \\text{Add 3}.\\hfill \\\\ x>1.5\\hfill & \\text{Divide by 2}.\\hfill \\end{array}[\/latex]<\/p>\n<p>In interval notation, the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x-3\\right)[\/latex] is [latex]\\left(1.5,\\infty \\right)[\/latex].<\/p>\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 argument 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\" style=\"text-align: center;\">[latex]\\begin{array}{c}x+3>0\\hfill & \\text{The argument 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 argument 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\" style=\"text-align: center;\">[latex]\\begin{array}{c}5 - 2x>0\\hfill & \\text{The argument 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\" 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, Logarithmic Model Showing Years as a Function of the Balance in the Account. The horizontal axis runs from 0 to 6000 and is labeled account balance. The vertical axis runs from 0 to 20 and is labeled years. The graph has an x-intercept at (2500, 0) and grows logarithmically. An arrow labels a point on the graph, at which the balance reaches 5000 dollars near year 14.\" 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 style=\"width: 70%;\" summary=\"Three rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td style=\"width: 30.3725%;\"><em><strong>x<\/strong><\/em><\/td>\n<td style=\"width: 15.6275%; text-align: center;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 11%; text-align: center;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 11%; text-align: center;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 6%; text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 10%; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 6%; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 1%; text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30.3725%;\"><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\n<td style=\"width: 15.6275%; text-align: center;\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<td style=\"width: 11%; text-align: center;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td style=\"width: 11%; text-align: center;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 6%; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 10%; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 6%; text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 1%; text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30.3725%;\"><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\n<td style=\"width: 15.6275%; text-align: center;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 11%; text-align: center;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 11%; text-align: center;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 6%; text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 10%; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 6%; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 1%; text-align: center;\">[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 style=\"width: 70%;\" 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>.\u00a0 This is a special property of the relationship between any function and its inverse.<\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Two side-by-side logarithmic graphs. The left graph is of the general form equation f of x equals log base b of x where b is greater than 1. The graph has a vertical asymptote of x equals 0 and points of (1, 0) and (b, 1). f of x approaches negative infinity as x approaches 0. The right graph is of the general form equation f of x equals log base b of x where b is between 0 and 1. The graph has a vertical asymptote of x equals 0 and points of (b, 1) and (1, 0). f of x approaches infinity as x approaches 0.\" width=\"824\" height=\"367\" \/><\/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\" style=\"text-align: left;\"><\/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-15992\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><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><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":13,"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.\"}]","CANDELA_OUTCOMES_GUID":"fa5f2606893c4c12b59edd1951325c62, f4efe9556537447b8fa3ef2374b1cf2d , 7a21435ce9a04f0aa6d0fcd6649ad18f ","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15992","chapter","type-chapter","status-publish","hentry"],"part":15981,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/15992","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/15992\/revisions"}],"predecessor-version":[{"id":20530,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/15992\/revisions\/20530"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/15981"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/15992\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=15992"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=15992"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=15992"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=15992"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}