{"id":5340,"date":"2021-10-13T18:24:25","date_gmt":"2021-10-13T18:24:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/chapter\/introduction-graphs-of-logarithmic-functions\/"},"modified":"2021-11-23T03:24:10","modified_gmt":"2021-11-23T03:24:10","slug":"introduction-graphs-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/chapter\/introduction-graphs-of-logarithmic-functions\/","title":{"raw":"Graphs of Logarithmic Functions","rendered":"Graphs of Logarithmic Functions"},"content":{"raw":"<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Characteristics of Graphs of Logarithmic Functions<\/span>\r\n\r\nBefore working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.\r\n\r\nRecall 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\r\n<ul>\r\n \t<li>The domain of <em>y<\/em>\u00a0is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n \t<li>The range of <em>y<\/em>\u00a0is [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\nIn 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:\r\n<ul>\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<h2>Graphing a Logarithmic Function Using a Table of Values<\/h2>\r\nNow that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions.\u00a0We begin with the 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 logarithmic form [latex]x={\\mathrm{log}}_{2}\\left(y\\right)[\/latex] in the table below.\r\n<table summary=\"Three rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 157px;\"><em><strong>x<\/strong><\/em><\/td>\r\n<td style=\"width: 221px;\">\u20133<\/td>\r\n<td style=\"width: 96px;\">\u20132<\/td>\r\n<td style=\"width: 96px;\">\u20131<\/td>\r\n<td style=\"width: 20px;\">0<\/td>\r\n<td style=\"width: 20px;\">1<\/td>\r\n<td style=\"width: 20px;\">2<\/td>\r\n<td style=\"width: 20px;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 157px;\"><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\r\n<td style=\"width: 221px;\">[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 96px;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 96px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 20px;\">1<\/td>\r\n<td style=\"width: 20px;\">2<\/td>\r\n<td style=\"width: 20px;\">4<\/td>\r\n<td style=\"width: 20px;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 157px;\"><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 221px;\">\u20133<\/td>\r\n<td style=\"width: 96px;\">\u20132<\/td>\r\n<td style=\"width: 96px;\">\u20131<\/td>\r\n<td style=\"width: 20px;\">0<\/td>\r\n<td style=\"width: 20px;\">1<\/td>\r\n<td style=\"width: 20px;\">2<\/td>\r\n<td style=\"width: 20px;\">3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing 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].\r\n<table style=\"height: 75px;\" summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\"><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n<td style=\"height: 30px;\">[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]\\left(0,1\\right)[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]\\left(1,2\\right)[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<td style=\"height: 30px;\">[latex]\\left(3,8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px;\"><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td style=\"height: 45px;\">[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\r\n<td style=\"height: 45px;\">[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\r\n<td style=\"height: 45px;\">[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\r\n<td style=\"height: 45px;\">[latex]\\left(1,0\\right)[\/latex]<\/td>\r\n<td style=\"height: 45px;\">[latex]\\left(2,1\\right)[\/latex]<\/td>\r\n<td style=\"height: 45px;\">[latex]\\left(4,2\\right)[\/latex]<\/td>\r\n<td style=\"height: 45px;\">[latex]\\left(8,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs we would expect, the <em>x\u00a0<\/em>and <em>y<\/em>-coordinates are reversed for the inverse functions. The figure below\u00a0shows the graphs of <em>f<\/em>\u00a0and <em>g<\/em>.\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\/896\/2016\/11\/02233818\/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.\" width=\"487\" height=\"438\" \/> 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 y\u00a0= x since they are inverses of each other.[\/caption]Observe the following from the graph:\r\n<ul>\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&nbsp;\r\n\r\nWatch the following video for an excellent demonstration on how to graph a logarithmic function...\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=w1A2ZYmfGco\r\n\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>A General Note: Characteristics of the Graph of the Function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nFor 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]:\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>one-to-one function<\/li>\r\n \t<li>vertical asymptote: <em>x\u00a0<\/em>= 0<\/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 0 &lt; <em>b\u00a0<\/em>&lt; 1<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>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>\r\n \t<li>If the function is in function notation, replace [latex]f\\left(x\\right)[\/latex] with <em>y<\/em>.<\/li>\r\n \t<li>Change the logarithmic equation to exponential form using the definition of the logarithm.<\/li>\r\n \t<li>Make a table of points, choosing the <em>y<\/em>-values first and finding the <em>x<\/em>-values, making sure to include the x-intercept.<\/li>\r\n \t<li>Plot the points. Make sure to include the <em>x<\/em>-intercept and the key point.<\/li>\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n \t<li>Draw and label the vertical asymptote.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Logarithmic Function Of the Form\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nGraph [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"909934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"909934\"]\r\n\r\nBefore graphing, identify the behavior and key points for the graph.\r\n<ul>\r\n \t<li>Since <em>b\u00a0<\/em>= 5 is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote <em>x\u00a0<\/em>= 0, 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 draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\r\n<\/ul>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233824\/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.\" width=\"487\" height=\"367\" \/> The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x = 0.[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGraph [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{5}}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"150661\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"150661\"]\r\n\r\nThe domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<span id=\"fs-id1165134377926\">\r\n<\/span>\r\n\r\n<img class=\"aligncenter size-full wp-image-3102\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16222857\/CNX_Precalc_Figure_04_04_0062.jpg\" alt=\"Graph of f(x)=log_(1\/5)(x) with labeled points at (1\/5, 1) and (1, 0). The y-axis is the asymptote.\" width=\"487\" height=\"367\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=34999&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\">\r\n<\/iframe>\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35000&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"250\">\r\n<\/iframe>\r\n\r\n<\/div>\r\nThe graphs below show 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] is base [latex]e\\approx \\text{2}.\\text{718}[\/latex] and [latex]\\mathrm{log} (x)[\/latex] is base 10.\r\n\r\n&nbsp;\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\/896\/2016\/11\/02233822\/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<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.<\/li>\r\n \t<li>Logarithmic equations can be written in an equivalent exponential form using the definition of a logarithm.<\/li>\r\n \t<li>Exponential equations can be written in an equivalent logarithmic form using the definition of a logarithm.<\/li>\r\n \t<li>Logarithmic functions with base <em>b<\/em>\u00a0can be evaluated mentally using previous knowledge of powers of <em>b<\/em>.<\/li>\r\n \t<li>Common logarithms can be evaluated mentally using previous knowledge of powers of 10.<\/li>\r\n \t<li>When common logarithms cannot be evaluated mentally, a calculator can be used.<\/li>\r\n \t<li>Natural logarithms can be evaluated using a calculator.<\/li>\r\n \t<li>To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for <em>x<\/em>.<\/li>\r\n \t<li>The graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] has an <em>x-<\/em>intercept at [latex]\\left(1,0\\right)[\/latex], domain [latex]\\left(0,\\infty \\right)[\/latex], range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], vertical asymptote <em>x\u00a0<\/em>= 0, and\r\n<ul>\r\n \t<li>if <em>b\u00a0<\/em>&gt; 1, the function is increasing.<\/li>\r\n \t<li>if 0 &lt; <em>b\u00a0<\/em>&lt; 1, the function is decreasing.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135397912\" class=\"definition\">\r\n \t<dt><strong>common logarithm<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135397918\">the exponent to which 10 must be raised to get <em>x<\/em>; [latex]{\\mathrm{log}}_{10}\\left(x\\right)[\/latex] is written simply as [latex]\\mathrm{log}\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135397926\" class=\"definition\">\r\n \t<dt><strong>logarithm<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135397932\">the exponent to which <em>b<\/em>\u00a0must be raised to get <em>x<\/em>; written [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137838635\" class=\"definition\">\r\n \t<dt><strong>natural logarithm<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137838640\">the exponent to which the number <em>e<\/em>\u00a0must be raised to get <em>x<\/em>; [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] is written as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>","rendered":"<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Characteristics of Graphs of Logarithmic Functions<\/span><\/p>\n<p>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>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>\n<li>The domain of <em>y<\/em>\u00a0is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<li>The range of <em>y<\/em>\u00a0is [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<p>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>\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<h2>Graphing a Logarithmic Function Using a Table of Values<\/h2>\n<p>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.\u00a0We begin with the 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 logarithmic form [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 style=\"width: 157px;\"><em><strong>x<\/strong><\/em><\/td>\n<td style=\"width: 221px;\">\u20133<\/td>\n<td style=\"width: 96px;\">\u20132<\/td>\n<td style=\"width: 96px;\">\u20131<\/td>\n<td style=\"width: 20px;\">0<\/td>\n<td style=\"width: 20px;\">1<\/td>\n<td style=\"width: 20px;\">2<\/td>\n<td style=\"width: 20px;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 157px;\"><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\n<td style=\"width: 221px;\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<td style=\"width: 96px;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td style=\"width: 96px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 20px;\">1<\/td>\n<td style=\"width: 20px;\">2<\/td>\n<td style=\"width: 20px;\">4<\/td>\n<td style=\"width: 20px;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 157px;\"><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\n<td style=\"width: 221px;\">\u20133<\/td>\n<td style=\"width: 96px;\">\u20132<\/td>\n<td style=\"width: 96px;\">\u20131<\/td>\n<td style=\"width: 20px;\">0<\/td>\n<td style=\"width: 20px;\">1<\/td>\n<td style=\"width: 20px;\">2<\/td>\n<td style=\"width: 20px;\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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=\"height: 75px;\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\"><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td style=\"height: 30px;\">[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]\\left(0,1\\right)[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]\\left(1,2\\right)[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td style=\"height: 30px;\">[latex]\\left(3,8\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px;\"><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td style=\"height: 45px;\">[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\n<td style=\"height: 45px;\">[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\n<td style=\"height: 45px;\">[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\n<td style=\"height: 45px;\">[latex]\\left(1,0\\right)[\/latex]<\/td>\n<td style=\"height: 45px;\">[latex]\\left(2,1\\right)[\/latex]<\/td>\n<td style=\"height: 45px;\">[latex]\\left(4,2\\right)[\/latex]<\/td>\n<td style=\"height: 45px;\">[latex]\\left(8,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As we would expect, the <em>x\u00a0<\/em>and <em>y<\/em>-coordinates are reversed for the inverse functions. The figure below\u00a0shows the graphs of <em>f<\/em>\u00a0and <em>g<\/em>.<\/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\/896\/2016\/11\/02233818\/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.\" width=\"487\" height=\"438\" \/><\/p>\n<p class=\"wp-caption-text\">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 y\u00a0= x since they are inverses of each other.<\/p>\n<\/div>\n<p>Observe the following from the graph:<\/p>\n<ul>\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<p>&nbsp;<\/p>\n<p>Watch the following video for an excellent demonstration on how to graph a logarithmic function&#8230;<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Graph an Exponential Function and Logarithmic Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/w1A2ZYmfGco?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Characteristics of the Graph of the Function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>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>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>one-to-one function<\/li>\n<li>vertical asymptote: <em>x\u00a0<\/em>= 0<\/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 0 &lt; <em>b\u00a0<\/em>&lt; 1<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>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>\n<li>If the function is in function notation, replace [latex]f\\left(x\\right)[\/latex] with <em>y<\/em>.<\/li>\n<li>Change the logarithmic equation to exponential form using the definition of the logarithm.<\/li>\n<li>Make a table of points, choosing the <em>y<\/em>-values first and finding the <em>x<\/em>-values, making sure to include the x-intercept.<\/li>\n<li>Plot the points. Make sure to include the <em>x<\/em>-intercept and the key point.<\/li>\n<li>Draw a smooth curve through the points.<\/li>\n<li>Draw and label the vertical asymptote.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Logarithmic Function Of the Form\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q909934\">Show Solution<\/span><\/p>\n<div id=\"q909934\" class=\"hidden-answer\" style=\"display: none\">\n<p>Before graphing, identify the behavior and key points for the graph.<\/p>\n<ul>\n<li>Since <em>b\u00a0<\/em>= 5 is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote <em>x\u00a0<\/em>= 0, 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 draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\n<\/ul>\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\/896\/2016\/11\/02233824\/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.\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x = 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{5}}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q150661\">Show Solution<\/span><\/p>\n<div id=\"q150661\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<span id=\"fs-id1165134377926\"><br \/>\n<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3102\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16222857\/CNX_Precalc_Figure_04_04_0062.jpg\" alt=\"Graph of f(x)=log_(1\/5)(x) with labeled points at (1\/5, 1) and (1, 0). The y-axis is the asymptote.\" width=\"487\" height=\"367\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=34999&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><br \/>\n<\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35000&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"250\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>The graphs below show 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] is base [latex]e\\approx \\text{2}.\\text{718}[\/latex] and [latex]\\mathrm{log} (x)[\/latex] is base 10.<\/p>\n<p>&nbsp;<\/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\/896\/2016\/11\/02233822\/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<h2>Key Concepts<\/h2>\n<ul>\n<li>The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.<\/li>\n<li>Logarithmic equations can be written in an equivalent exponential form using the definition of a logarithm.<\/li>\n<li>Exponential equations can be written in an equivalent logarithmic form using the definition of a logarithm.<\/li>\n<li>Logarithmic functions with base <em>b<\/em>\u00a0can be evaluated mentally using previous knowledge of powers of <em>b<\/em>.<\/li>\n<li>Common logarithms can be evaluated mentally using previous knowledge of powers of 10.<\/li>\n<li>When common logarithms cannot be evaluated mentally, a calculator can be used.<\/li>\n<li>Natural logarithms can be evaluated using a calculator.<\/li>\n<li>To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for <em>x<\/em>.<\/li>\n<li>The graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] has an <em>x-<\/em>intercept at [latex]\\left(1,0\\right)[\/latex], domain [latex]\\left(0,\\infty \\right)[\/latex], range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], vertical asymptote <em>x\u00a0<\/em>= 0, and\n<ul>\n<li>if <em>b\u00a0<\/em>&gt; 1, the function is increasing.<\/li>\n<li>if 0 &lt; <em>b\u00a0<\/em>&lt; 1, the function is decreasing.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135397912\" class=\"definition\">\n<dt><strong>common logarithm<\/strong><\/dt>\n<dd id=\"fs-id1165135397918\">the exponent to which 10 must be raised to get <em>x<\/em>; [latex]{\\mathrm{log}}_{10}\\left(x\\right)[\/latex] is written simply as [latex]\\mathrm{log}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135397926\" class=\"definition\">\n<dt><strong>logarithm<\/strong><\/dt>\n<dd id=\"fs-id1165135397932\">the exponent to which <em>b<\/em>\u00a0must be raised to get <em>x<\/em>; written [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137838635\" class=\"definition\">\n<dt><strong>natural logarithm<\/strong><\/dt>\n<dd id=\"fs-id1165137838640\">the exponent to which the number <em>e<\/em>\u00a0must be raised to get <em>x<\/em>; [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] is written as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\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-5340\">\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>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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Questoin ID 34999, 35000. <strong>Authored by<\/strong>: Smart, Jim. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 74340, 74341. <strong>Authored by<\/strong>: Nearing, Daniel. <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>: IMathAS Community License CC-BY + GPL<\/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":167848,"menu_order":6,"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\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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