{"id":100,"date":"2021-06-04T18:08:57","date_gmt":"2021-06-04T18:08:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/chapter\/graphs-of-logarithmic-functions\/"},"modified":"2021-06-14T23:52:19","modified_gmt":"2021-06-14T23:52:19","slug":"graphs-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/chapter\/graphs-of-logarithmic-functions\/","title":{"raw":"5.4 Graphs of Logarithmic Functions","rendered":"5.4 Graphs of 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-id1165135194555\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/introduction-to-graphs-of-exponential-functions\/\" target=\"_blank\" rel=\"noopener\">Graphs of Exponential Functions<\/a>, we saw how creating a graphical representation of an exponential model gives 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 5%, 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? Figure 1\u00a0shows 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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010821\/CNX_Precalc_Figure_04_04_0012.jpg\" alt=\"A graph titled, \" width=\"900\" height=\"459\" \/> <b>Figure 1<\/b>[\/caption]\r\n<p id=\"fs-id1165135161452\">In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.<\/p>\r\n\r\n<h2>Identify the domain of a logarithmic function<\/h2>\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 <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\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<p id=\"fs-id1165135245571\">Transformations of the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections\u2014to the parent function without loss of shape.<\/p>\r\n<p id=\"fs-id1165137653624\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/introduction-to-graphs-of-exponential-functions\/\" target=\"_blank\" rel=\"noopener\">Graphs of Exponential Functions<\/a> we saw that certain transformations can change the <em>range<\/em> of [latex]y={b}^{x}[\/latex]. Similarly, applying transformations to the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] can change the <em>domain<\/em>. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists <em>only of positive real numbers<\/em>. That is, the argument of the logarithmic function must be greater than zero.<\/p>\r\n<p id=\"fs-id1165137851584\">For example, consider [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x - 3\\right)[\/latex]. This function is defined for any values of <em>x<\/em>\u00a0such that the argument, in this case [latex]2x - 3[\/latex], is greater than zero. To find the domain, we set up an inequality and solve for\u00a0<em>x<\/em>:<\/p>\r\n\r\n<div id=\"eip-318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&amp;2x - 3&gt;0 &amp;&amp; \\text{Show the argument greater than zero}. \\\\ &amp;2x&gt;3 &amp;&amp; \\text{Add 3}. \\\\ &amp;x&gt;1.5 &amp;&amp; \\text{Divide by 2}. \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137645047\">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>\r\n\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\n<div id=\"Example_04_04_01\" class=\"example\">\r\n<div id=\"fs-id1165137846475\" class=\"exercise\">\r\n<div id=\"fs-id1165137460694\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Identifying the Domain of a Logarithmic Shift<\/h3>\r\n<p id=\"fs-id1165135209576\">What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex]?<\/p>\r\n[reveal-answer q=\"174870\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"174870\"]\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<p style=\"text-align: center;\">[latex]\\begin{align}&amp;x+3&gt;0 &amp;&amp; \\text{The input must be positive}. \\\\ &amp;x&gt;-3 &amp;&amp; \\text{Subtract 3}. \\end{align}[\/latex]<\/p>\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\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137645484\">What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x - 2\\right)+1[\/latex]?<\/p>\r\n[reveal-answer q=\"405290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"405290\"]\r\n\r\n[latex]\\left(2,\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_04_04_02\" class=\"example\">\r\n<div id=\"fs-id1165137894615\" class=\"exercise\">\r\n<div id=\"fs-id1165134108527\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Identifying the Domain of a Logarithmic Shift and Reflection<\/h3>\r\n<p id=\"fs-id1165135499558\">What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex]?<\/p>\r\n[reveal-answer q=\"675604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"675604\"]\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<p style=\"text-align: center;\">[latex]\\begin{align}&amp;5 - 2x&gt;0 &amp;&amp; \\text{The input must be positive}. \\\\ &amp;-2x&gt;-5 &amp;&amp; \\text{Subtract }5. \\\\ &amp;x&lt;\\frac{5}{2} &amp;&amp; \\text{Divide by }-2\\text{ and switch the inequality}. \\end{align}[\/latex]<\/p>\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<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137453336\">What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(x - 5\\right)+2[\/latex]?<\/p>\r\n[reveal-answer q=\"87516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"87516\"]\r\n\r\n[latex]\\left(5,\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174284[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>\u00a0Graph logarithmic functions<\/h2>\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 with 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 id=\"Table_04_04_01\" 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>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/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>1<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>8<\/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>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/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 id=\"Table_04_04_02\" 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\u2019d 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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010821\/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\n<p style=\"text-align: center;\"><strong>Figure 2.\u00a0<\/strong>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>\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<div id=\"fs-id1165137780760\" class=\"note textbox\">\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>= 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<figure id=\"CNX_Precalc_Figure_04_04_003\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010821\/CNX_Precalc_Figure_04_04_003G2.jpg\" alt=\"&quot;Two\" \/><\/figure>\r\nFigure 3\u00a0shows 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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/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\" \/> <strong>Figure 4.\u00a0<\/strong>The graphs of three logarithmic functions with different bases, all greater than 1.[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137871937\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137805513\">How To: Given a logarithmic function with 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>Draw and label the vertical asymptote, <em>x<\/em> = 0.<\/li>\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], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote, <em>x<\/em> = 0.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_03\" class=\"example\">\r\n<div id=\"fs-id1165137550508\" class=\"exercise\">\r\n<div id=\"fs-id1165137550510\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Graphing a Logarithmic Function with the Form\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex].<\/h3>\r\n<p id=\"fs-id1165137431970\">Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"347847\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"347847\"]\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 <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<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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/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=\"557\" height=\"419\" \/><\/span><\/figure>\r\n<p id=\"fs-id1165135697920\" style=\"text-align: center;\"><strong>Figure 5.\u00a0<\/strong>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<\/em> = 0.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135171582\">Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{5}}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"808887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808887\"]\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<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/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.\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174289[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>\u00a0Graphing Transformations of Logarithmic Functions<\/h2>\r\n<p id=\"fs-id1165137430986\">As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the <strong>parent function<\/strong> [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] without loss of shape.<\/p>\r\n\r\n<section id=\"fs-id1165137734884\">\r\n<h2>Graphing a Horizontal Shift of\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h2>\r\nWhen a constant <em>c<\/em>\u00a0is added to the input of the parent function [latex]f\\left(x\\right)=\\text{log}_{b}\\left(x\\right)[\/latex], the result is a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em>opposite<\/em> direction of the sign on <em>c<\/em>. To visualize horizontal shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] and for <em>c\u00a0<\/em>&gt; 0 alongside the shift left, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], and the shift right, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/CNX_Precalc_Figure_04_04_007n2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.\" width=\"900\" height=\"526\" \/> <b>Figure 6<\/b>[\/caption]\r\n\r\n<div id=\"fs-id1165135296307\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Horizontal Shifts of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\n<p id=\"fs-id1165135176174\">For any constant <em>c<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165135206192\">\r\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\r\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\r\n \t<li>has the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\r\n \t<li>has domain [latex]\\left(-c,\\infty \\right)[\/latex].<\/li>\r\n \t<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137641710\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137641715\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], graph the translation.<\/h3>\r\n<ol id=\"fs-id1165137454284\">\r\n \t<li>Identify the horizontal shift:\r\n<ol id=\"fs-id1165137454288\">\r\n \t<li>If <em>c<\/em> &gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>\u00a0units.<\/li>\r\n \t<li>If <em>c\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>\u00a0units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting <em>c<\/em>\u00a0from the\u00a0<em>x<\/em>\u00a0coordinate.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The Domain is [latex]\\left(-c,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u2013c.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_04\" class=\"example\">\r\n<div id=\"fs-id1165137414959\" class=\"exercise\">\r\n<div id=\"fs-id1165137414961\" class=\"problem textbox shaded\">\r\n<h3>Example 4:\u00a0Graphing a Horizontal Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\n<p id=\"fs-id1165137455420\">Sketch the horizontal shift [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"785817\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"785817\"]\r\n<p id=\"fs-id1165137759885\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex], we notice [latex]x+\\left(-2\\right)=x - 2[\/latex].<\/p>\r\n<p id=\"fs-id1165137784630\">Thus <em>c\u00a0<\/em>= \u20132, so <em>c\u00a0<\/em>&lt; 0. This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] right 2 units.<\/p>\r\n<p id=\"fs-id1165137836995\">The vertical asymptote is [latex]x=-\\left(-2\\right)[\/latex] or <em>x\u00a0<\/em>= 2.<\/p>\r\n<p id=\"fs-id1165134042608\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165137475806\">The new coordinates are found by adding 2 to the <em>x<\/em>\u00a0coordinates.<\/p>\r\n<p id=\"fs-id1165137748449\">Label the points [latex]\\left(\\frac{7}{3},-1\\right)[\/latex], [latex]\\left(3,0\\right)[\/latex], and [latex]\\left(5,1\\right)[\/latex].<\/p>\r\nThe domain is [latex]\\left(2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010823\/CNX_Precalc_Figure_04_04_0082.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).\" width=\"487\" height=\"363\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<span style=\"font-size: 0.9em;\">\u00a0<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135329937\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"139882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"139882\"]\r\n\r\nThe domain is [latex]\\left(-4,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the asymptote <em>x\u00a0<\/em>= \u20134.<span id=\"fs-id1165135209395\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010823\/CNX_Precalc_Figure_04_04_0092.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174300[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing a Vertical Shift of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/span>\r\n\r\n<\/section><section id=\"fs-id1165135403538\">When a constant <em>d<\/em>\u00a0is added to the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], the result is a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the direction of the sign on <em>d<\/em>. To visualize vertical shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift up, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] and the shift down, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010823\/CNX_Precalc_Figure_04_04_010F2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.\" width=\"900\" height=\"684\" \/> <b>Figure 8<\/b>[\/caption]\r\n\r\n<div id=\"fs-id1165137767601\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Vertical Shifts of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\n<p id=\"fs-id1165137661370\">For any constant <em>d<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137803105\">\r\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\r\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\r\n \t<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n \t<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137706002\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137706009\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex], graph the translation.<\/h3>\r\n<ol>\r\n \t<li>Identify the vertical shift:\r\n<ol>\r\n \t<li>If <em>d\u00a0<\/em>&gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>\u00a0units.<\/li>\r\n \t<li>If <em>d\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d\u00a0<\/em>units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding <em>d<\/em>\u00a0to the <em>y\u00a0<\/em>coordinate.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>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.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_05\" class=\"example\">\r\n<div id=\"fs-id1165137470057\" class=\"exercise\">\r\n<div id=\"fs-id1165137470059\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Graphing a Vertical Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\n<p id=\"fs-id1165137832038\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"657475\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"657475\"]\r\n<p id=\"fs-id1165137465913\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex], we will notice <em>d\u00a0<\/em>= \u20132. Thus <em>d\u00a0<\/em>&lt; 0.<\/p>\r\n<p id=\"fs-id1165135175015\">This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] down 2 units.<\/p>\r\n<p id=\"fs-id1165137644429\">The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\r\n<p id=\"fs-id1165137408419\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165135503945\">The new coordinates are found by subtracting 2 from the <em>y <\/em>coordinates.<\/p>\r\n<p id=\"fs-id1165135421660\">Label the points [latex]\\left(\\frac{1}{3},-3\\right)[\/latex], [latex]\\left(1,-2\\right)[\/latex], and [latex]\\left(3,-1\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165135195524\">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<\/em> = 0.<span id=\"fs-id1165134393856\">\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010823\/CNX_Precalc_Figure_04_04_0112.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 1).\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137698285\" style=\"text-align: center;\"><strong>Figure 9.\u00a0<\/strong>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<\/em> = 0.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137760886\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)+2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"597513\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"597513\"]\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-id1165137874471\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_0122.jpg\" alt=\"Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174304[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing Stretches and Compressions of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/span>\r\n\r\n<\/section><section id=\"fs-id1165137770245\">When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by a constant <em>a<\/em> &gt; 0, the result is a <strong>vertical stretch<\/strong> or <strong>compression<\/strong> of the original graph. To visualize stretches and compressions, we set <em>a\u00a0<\/em>&gt; 1 and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the vertical stretch, [latex]g\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] and the vertical compression, [latex]h\\left(x\\right)=\\frac{1}{a}{\\mathrm{log}}_{b}\\left(x\\right)[\/latex].<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_013n2.jpg\" alt=\"&quot;Graph\" \/>\r\n<div id=\"fs-id1165137433996\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Vertical Stretches and Compressions of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\n<p id=\"fs-id1165137758179\">For any constant <em>a<\/em> &gt; 1, the function [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137428102\">\r\n \t<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if <em>a\u00a0<\/em>&gt; 1.<\/li>\r\n \t<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if 0 &lt; <em>a\u00a0<\/em>&lt; 1.<\/li>\r\n \t<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>has the <em>x<\/em>-intercept [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n \t<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135169301\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135169307\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], [latex]a&gt;0[\/latex], graph the translation.<\/h3>\r\n<ol id=\"fs-id1165137464127\">\r\n \t<li>Identify the vertical stretch or compressions:\r\n<ol id=\"eip-id1165134081434\">\r\n \t<li>If [latex]|a|&gt;1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is stretched by a factor of <em>a<\/em>\u00a0units.<\/li>\r\n \t<li>If [latex]|a|&lt;1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is compressed by a factor of <em>a<\/em>\u00a0units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the <em>y<\/em>\u00a0coordinates by <em>a<\/em>.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>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<\/em> = 0.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_06\" class=\"example\">\r\n<div id=\"fs-id1165135309914\" class=\"exercise\">\r\n<div id=\"fs-id1165135309916\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Graphing a Stretch or Compression of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\n<p id=\"fs-id1165137602128\">Sketch a graph of [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"846570\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"846570\"]\r\n<p id=\"fs-id1165135210052\">Since the function is [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex], we will notice <em>a\u00a0<\/em>= 2.<\/p>\r\n<p id=\"fs-id1165135384321\">This means we will stretch the function [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] by a factor of 2.<\/p>\r\n<p id=\"fs-id1165135481989\">The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\r\n<p id=\"fs-id1165137757801\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{4},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(4,1\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165135570058\">The new coordinates are found by multiplying the <em>y<\/em>\u00a0coordinates by 2.<\/p>\r\n<p id=\"fs-id1165137837989\">Label the points [latex]\\left(\\frac{1}{4},-2\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(4,\\text{2}\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165135543469\">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-id1165134059742\">\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_0142.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).\" \/><\/span><\/p>\r\n<p id=\"fs-id1165135566827\" style=\"text-align: center;\"><strong>Figure 11.\u00a0<\/strong>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.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135471122\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"645251\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"645251\"]\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-id1165135332505\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_0152.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1\/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_04_04_07\" class=\"example\">\r\n<div id=\"fs-id1165134267814\" class=\"exercise\">\r\n<div id=\"fs-id1165134267816\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Combining a Shift and a Stretch<\/h3>\r\n<p id=\"fs-id1165137863045\">Sketch a graph of [latex]f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"470378\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"470378\"]\r\n<p id=\"fs-id1165137935561\">Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5. The vertical asymptote will be shifted to <em>x\u00a0<\/em>= \u20132. The <em>x<\/em>-intercept will be [latex]\\left(-1,0\\right)[\/latex]. The domain will be [latex]\\left(-2,\\infty \\right)[\/latex]. Two points will help give the shape of the graph: [latex]\\left(-1,0\\right)[\/latex] and [latex]\\left(8,5\\right)[\/latex]. We chose <em>x\u00a0<\/em>= 8 as the <em>x<\/em>-coordinate of one point to graph because when <em>x\u00a0<\/em>= 8, <em>x\u00a0<\/em>+ 2 = 10, the base of the common logarithm.<span id=\"fs-id1165135641650\">\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_0162.jpg\" alt=\"Graph of three functions. The parent function is y=log(x), with an asymptote at x=0. The first translation function y=5log(x+2) has an asymptote at x=-2. The second translation function y=log(x+2) has an asymptote at x=-2.\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137874883\" style=\"text-align: center;\"><strong>Figure 12.\u00a0<\/strong>The domain is [latex]\\left(-2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u20132.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137838697\">Sketch a graph of the function [latex]f\\left(x\\right)=3\\mathrm{log}\\left(x - 2\\right)+1[\/latex]. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"793205\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"793205\"]\r\n\r\nThe domain is [latex]\\left(2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.\r\n<div id=\"fs-id1165137437228\" class=\"solution\">\r\n\r\n<span id=\"fs-id1165135177663\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010825\/CNX_Precalc_Figure_04_04_0172.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" \/><\/span>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174299[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing Reflections of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/span>\r\n\r\n<\/section><section id=\"fs-id1165137629003\">\r\n<p id=\"fs-id1165135169315\">When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a <strong>reflection<\/strong> about the <em>x<\/em>-axis. When the <em>input<\/em> is multiplied by \u20131, the result is a reflection about the <em>y<\/em>-axis. To visualize reflections, we restrict <em>b\u00a0<\/em>&gt; 1, and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex] and the reflection about the <em>y<\/em>-axis, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex].<\/p>\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010825\/CNX_Precalc_Figure_04_04_018n2.jpg\" alt=\"&quot;Graph\" \/>\r\n<div id=\"fs-id1165135190744\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Reflections of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\n<p id=\"fs-id1165137722409\">The function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137832285\">\r\n \t<li>reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis.<\/li>\r\n \t<li>has domain, [latex]\\left(0,\\infty \\right)[\/latex], range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and vertical asymptote, <em>x\u00a0<\/em>= 0, which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\nThe function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]\r\n<ul id=\"fs-id1165137734930\">\r\n \t<li>reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y<\/em>-axis.<\/li>\r\n \t<li>has domain [latex]\\left(-\\infty ,0\\right)[\/latex].<\/li>\r\n \t<li>has range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and vertical asymptote, <em>x\u00a0<\/em>= 0, which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137638830\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137638837\">How To: Given a logarithmic function with the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph a translation.<\/h3>\r\n<table id=\"Table_04_04_08\" class=\"unnumbered\" summary=\"The first column gives the following instructions of graphing a translation of f(x)=-log_b(x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the x-axis; 4. Draw a smooth curve through the points; 5. State the domain, (0, infinity), the range, (-infinity, infinity), and the vertical asymptote x=0. The second column gives the following instructions of graphing a translation of f(x)=log_b(-x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (-1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the y-axis; 4. Draw a smooth curve through the points; 5. State the domain, (-infinity, 0), the range, (-infinity, infinity), and the vertical asymptote x=0.\">\r\n<thead>\r\n<tr>\r\n<th>[latex]\\text{If }f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n<th>[latex]\\text{If }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1. Draw the vertical asymptote, <em>x\u00a0<\/em>= 0.<\/td>\r\n<td>1. Draw the vertical asymptote, <em>x\u00a0<\/em>= 0.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\r\n<td>2. Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis.<\/td>\r\n<td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y<\/em>-axis.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. Draw a smooth curve through the points.<\/td>\r\n<td>4. Draw a smooth curve through the points.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5. State the domain, [latex]\\left(0,\\infty \\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\r\n<td>5. State the domain, [latex]\\left(-\\infty ,0\\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"Example_04_04_08\" class=\"example\">\r\n<div id=\"fs-id1165137697928\" class=\"exercise\">\r\n<div id=\"fs-id1165137849033\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Graphing a Reflection of a Logarithmic Function<\/h3>\r\n<p id=\"fs-id1165137849038\">Sketch a graph of [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"618451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"618451\"]\r\n<p id=\"fs-id1165137836525\">Before graphing [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex], identify the behavior and key points for the graph.<\/p>\r\n\r\n<ul id=\"fs-id1165137769879\">\r\n \t<li>Since <em>b\u00a0<\/em>= 10 is greater than one, we know that the parent function is increasing. Since the <em>input<\/em> value is multiplied by \u20131, <em>f<\/em>\u00a0is a reflection of the parent graph about the <em>y-<\/em>axis. Thus, [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] will be decreasing as <em>x<\/em>\u00a0moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>The <em>x<\/em>-intercept is [latex]\\left(-1,0\\right)[\/latex].<\/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<figure id=\"CNX_Precalc_Figure_04_04_019\" class=\"small\"><span id=\"fs-id1165134042188\"> <img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010825\/CNX_Precalc_Figure_04_04_0192.jpg\" alt=\"Graph of two functions. The parent function is y=log(x), with an asymptote at x=0 and labeled points at (1, 0), and (10, 0).The translation function f(x)=log(-x) has an asymptote at x=0 and labeled points at (-1, 0) and (-10, 1).\" \/><\/span><\/figure>\r\n<p id=\"fs-id1165134042202\" style=\"text-align: center;\"><strong>Figure 14.\u00a0<\/strong>The domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135681852\">Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"485222\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"485222\"]\r\n\r\nThe domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<span id=\"fs-id1165137855148\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010825\/CNX_Precalc_Figure_04_04_0202.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134579621\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165134579627\">How To: Given a logarithmic equation, use a graphing calculator to approximate solutions.<\/h3>\r\n<ol id=\"fs-id1165137431118\">\r\n \t<li>Press <strong>[Y=]<\/strong>. Enter the given logarithm equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\r\n \t<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\r\n \t<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \"intersect\" and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x<\/em>, for the point(s) of intersection.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_09\" class=\"example\">\r\n<div id=\"fs-id1165135414229\" class=\"exercise\">\r\n<div id=\"fs-id1165135414231\" class=\"problem textbox shaded\">\r\n<h3>Example 9: Approximating the Solution of a Logarithmic Equation<\/h3>\r\n<p id=\"fs-id1165135414236\">Solve [latex]4\\mathrm{ln}\\left(x\\right)+1=-2\\mathrm{ln}\\left(x - 1\\right)[\/latex] graphically. Round to the nearest thousandth.<\/p>\r\n[reveal-answer q=\"513609\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"513609\"]\r\n<p id=\"fs-id1165135193434\">Press <strong>[Y=]<\/strong> and enter [latex]4\\mathrm{ln}\\left(x\\right)+1[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter [latex]-2\\mathrm{ln}\\left(x - 1\\right)[\/latex] next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for <em>x<\/em>\u00a0and \u201310 to 10 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to right of <em>x\u00a0<\/em>= 1.<\/p>\r\n<p id=\"fs-id1165135245763\">For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em>x<\/em>-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) So, to the nearest thousandth, [latex]x\\approx 1.339[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137639531\">Solve [latex]5\\mathrm{log}\\left(x+2\\right)=4-\\mathrm{log}\\left(x\\right)[\/latex] graphically. Round to the nearest thousandth.<\/p>\r\n[reveal-answer q=\"862673\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"862673\"]\r\n\r\n[latex]x\\approx 3.049[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135528930\">\r\n<h2>Summarizing Translations of the Logarithmic Function<\/h2>\r\n<p id=\"fs-id1165135528935\">Now that we have worked with each type of translation for the logarithmic function, we can summarize each in the table below\u00a0to arrive at the general equation for translating exponential functions.<\/p>\r\n\r\n<table id=\"Table_04_04_009\" summary=\"Titled, \">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Translations of the Parent Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th style=\"text-align: center;\">Translation<\/th>\r\n<th style=\"text-align: center;\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Shift\r\n<ul id=\"fs-id1165137416971\">\r\n \t<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\r\n \t<li>Vertically <em>d<\/em>\u00a0units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Stretch and Compress\r\n<ul id=\"fs-id1165137427553\">\r\n \t<li>Stretch if [latex]|a|&gt;1[\/latex]<\/li>\r\n \t<li>Compression if [latex]|a|&lt;1[\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em>x<\/em>-axis<\/td>\r\n<td>[latex]y=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em>y<\/em>-axis<\/td>\r\n<td>[latex]y={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>General equation for all translations<\/td>\r\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1165137414493\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Translations of Logarithmic Functions<\/h3>\r\n<p id=\"fs-id1165137414501\">All translations of the parent logarithmic function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], have the form<\/p>\r\n\r\n<div id=\"fs-id1165135408512\" class=\"equation\" style=\"text-align: center;\">[latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/div>\r\n<p id=\"fs-id1165137734655\">where the parent function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right),b&gt;1[\/latex], is<\/p>\r\n\r\n<ul id=\"fs-id1165137531610\">\r\n \t<li>shifted vertically up <em>d<\/em>\u00a0units.<\/li>\r\n \t<li>shifted horizontally to the left <em>c<\/em>\u00a0units.<\/li>\r\n \t<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\r\n \t<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\r\n \t<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137725084\">For [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex], the graph of the parent function is reflected about the <em>y<\/em>-axis.<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_04_04_10\" class=\"example\">\r\n<div id=\"fs-id1165135296269\" class=\"exercise\">\r\n<div id=\"fs-id1165135296271\" class=\"problem textbox shaded\">\r\n<h3>Example 10: Finding the Vertical Asymptote of a Logarithm Graph<\/h3>\r\n<p id=\"fs-id1165135296276\">What is the vertical asymptote of [latex]f\\left(x\\right)=-2{\\mathrm{log}}_{3}\\left(x+4\\right)+5[\/latex]?<\/p>\r\n[reveal-answer q=\"841705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841705\"]\r\n\r\nThe vertical asymptote is at <em>x\u00a0<\/em>= \u20134.\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137871960\">The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to <em>x\u00a0<\/em>= \u20134.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135368433\">What is the vertical asymptote of [latex]f\\left(x\\right)=3+\\mathrm{ln}\\left(x - 1\\right)[\/latex]?<\/p>\r\n[reveal-answer q=\"975284\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"975284\"]\r\n\r\n<em>x\u00a0<\/em>= 1\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_04_04_11\" class=\"example\">\r\n<div id=\"fs-id1165137849555\" class=\"exercise\">\r\n<div id=\"fs-id1165137849558\" class=\"problem textbox shaded\">\r\n<h3>Example 11: Finding the Equation from a Graph<\/h3>\r\n<p id=\"fs-id1165137849563\">Find a possible equation for the common logarithmic function graphed in Figure 15.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005323\/CNX_Precalc_Figure_04_04_021.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).\" width=\"487\" height=\"367\" \/> <b>Figure 15<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"993624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"993624\"]\r\n<p id=\"fs-id1165135342979\">This graph has a vertical asymptote at <em>x\u00a0<\/em>= \u20132 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+k[\/latex]<\/p>\r\n<p id=\"fs-id1165135406913\">It appears the graph passes through the points [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(2,-1\\right)[\/latex]. Substituting [latex]\\left(-1,1\\right)[\/latex],<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;1=-a\\mathrm{log}\\left(-1+2\\right)+k &amp;&amp; \\text{Substitute }\\left(-1,1\\right). \\\\ &amp;1=-a\\mathrm{log}\\left(1\\right)+k &amp;&amp; \\text{Arithmetic}. \\\\ &amp;1=k &amp;&amp; \\text{log(1)}=0. \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137628655\">Next, substituting in [latex]\\left(2,-1\\right)[\/latex],<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;-1=-a\\mathrm{log}\\left(2+2\\right)+1 &amp;&amp; \\text{Plug in }\\left(2,-1\\right). \\\\ &amp;-2=-a\\mathrm{log}\\left(4\\right) &amp;&amp; \\text{Arithmetic}. \\\\ &amp;a=\\frac{2}{\\mathrm{log}\\left(4\\right)}&amp;&amp; \\text{Solve for }a. \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135192211\">This gives us the equation [latex]f\\left(x\\right)=-\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137735586\">We can verify this answer by comparing the function values in the table below\u00a0with the points on the graph in Example 11.<\/p>\r\n\r\n<table id=\"Table_04_04_010\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<td>\u22120.58496<\/td>\r\n<td>\u22121<\/td>\r\n<td>\u22121.3219<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\r\n<td>\u22121.5850<\/td>\r\n<td>\u22121.8074<\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22122.1699<\/td>\r\n<td>\u22122.3219<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137665487\">Give the equation of the natural logarithm graphed in Figure 16.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005324\/CNX_Precalc_Figure_04_04_022.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).\" width=\"487\" height=\"442\" \/> <b>Figure 16<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"537017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"537017\"]\r\n\r\n[latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x+3\\right)-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137855236\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165137855242\"><strong>Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?<\/strong><\/p>\r\n<p id=\"fs-id1165137827126\"><em>Yes, if we know the function is a general logarithmic function. For example, look at the graph in Try It 11. The graph approaches x = \u20133 (or thereabouts) more and more closely, so x = \u20133 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex]\\left\\{x|x&gt;-3\\right\\}[\/latex]. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as [latex]x\\to -{3}^{+},f\\left(x\\right)\\to -\\infty [\/latex] and as [latex]x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex].<\/em><\/p>\r\n\r\n<\/div>\r\n<\/section><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span>\r\n\r\n<section id=\"fs-id1165137749167\" class=\"key-equations\">\r\n<table id=\"fs-id1737642\" summary=\"...\">\r\n<tbody>\r\n<tr>\r\n<td>General Form for the Translation of the Parent Logarithmic Function [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<td>[latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165137863125\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165137863132\">\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 id=\"fs-id1165135441773\">\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 \t<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] horizontally\r\n<ul id=\"fs-id1165135512562\">\r\n \t<li>left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\r\n \t<li>right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically\r\n<ul id=\"fs-id1165137761068\">\r\n \t<li>up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\r\n \t<li>down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>For any constant <em>a\u00a0<\/em>&gt; 0, the equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\r\n<ul id=\"fs-id1165134040579\">\r\n \t<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if |<em>a<\/em>| &gt; 1.<\/li>\r\n \t<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if |<em>a<\/em>| &lt; 1.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>When the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a reflection about the <em>x<\/em>-axis. When the input is multiplied by \u20131, the result is a reflection about the <em>y<\/em>-axis.\r\n<ul id=\"fs-id1165135186594\">\r\n \t<li>The equation [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] represents a reflection of the parent function about the <em>x-<\/em>axis.<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex] represents a reflection of the parent function about the <em>y-<\/em>axis.<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1165137834414\">\r\n \t<li>A graphing calculator may be used to approximate solutions to some logarithmic equations.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>All translations of the logarithmic function can be summarized by the general equation [latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex].<\/li>\r\n \t<li>Given an equation with the general form [latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can identify the vertical asymptote <em>x\u00a0<\/em>= \u2013c for the transformation.<\/li>\r\n \t<li>Using the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can write the equation of a logarithmic function given its graph.<\/li>\r\n<\/ul>\r\n<\/section>","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-id1165135194555\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/introduction-to-graphs-of-exponential-functions\/\" target=\"_blank\" rel=\"noopener\">Graphs of Exponential Functions<\/a>, we saw how creating a graphical representation of an exponential model gives 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 5%, 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? Figure 1\u00a0shows 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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010821\/CNX_Precalc_Figure_04_04_0012.jpg\" alt=\"A graph titled,\" width=\"900\" height=\"459\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135161452\">In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.<\/p>\n<h2>Identify the domain of a logarithmic function<\/h2>\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 <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 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 id=\"fs-id1165135245571\">Transformations of the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections\u2014to the parent function without loss of shape.<\/p>\n<p id=\"fs-id1165137653624\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/introduction-to-graphs-of-exponential-functions\/\" target=\"_blank\" rel=\"noopener\">Graphs of Exponential Functions<\/a> we saw that certain transformations can change the <em>range<\/em> of [latex]y={b}^{x}[\/latex]. Similarly, applying transformations to the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] can change the <em>domain<\/em>. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists <em>only of positive real numbers<\/em>. That is, the argument of the logarithmic function must be greater than zero.<\/p>\n<p id=\"fs-id1165137851584\">For example, consider [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x - 3\\right)[\/latex]. This function is defined for any values of <em>x<\/em>\u00a0such that the argument, in this case [latex]2x - 3[\/latex], is greater than zero. To find the domain, we set up an inequality and solve for\u00a0<em>x<\/em>:<\/p>\n<div id=\"eip-318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&2x - 3>0 && \\text{Show the argument greater than zero}. \\\\ &2x>3 && \\text{Add 3}. \\\\ &x>1.5 && \\text{Divide by 2}. \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137645047\">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<div id=\"Example_04_04_01\" class=\"example\">\n<div id=\"fs-id1165137846475\" class=\"exercise\">\n<div id=\"fs-id1165137460694\" class=\"problem textbox shaded\">\n<h3>Example 1: Identifying the Domain of a Logarithmic Shift<\/h3>\n<p id=\"fs-id1165135209576\">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=\"q174870\">Show Solution<\/span><\/p>\n<div id=\"q174870\" 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<p style=\"text-align: center;\">[latex]\\begin{align}&x+3>0 && \\text{The input must be positive}. \\\\ &x>-3 && \\text{Subtract 3}. \\end{align}[\/latex]<\/p>\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<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137645484\">What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x - 2\\right)+1[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q405290\">Show Solution<\/span><\/p>\n<div id=\"q405290\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(2,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_04_04_02\" class=\"example\">\n<div id=\"fs-id1165137894615\" class=\"exercise\">\n<div id=\"fs-id1165134108527\" class=\"problem textbox shaded\">\n<h3>Example 2: Identifying the Domain of a Logarithmic Shift and Reflection<\/h3>\n<p id=\"fs-id1165135499558\">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=\"q675604\">Show Solution<\/span><\/p>\n<div id=\"q675604\" 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<p style=\"text-align: center;\">[latex]\\begin{align}&5 - 2x>0 && \\text{The input must be positive}. \\\\ &-2x>-5 && \\text{Subtract }5. \\\\ &x<\\frac{5}{2} && \\text{Divide by }-2\\text{ and switch the inequality}. \\end{align}[\/latex]<\/p>\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<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137453336\">What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(x - 5\\right)+2[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q87516\">Show Solution<\/span><\/p>\n<div id=\"q87516\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(5,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174284\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174284&theme=oea&iframe_resize_id=ohm174284\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>\u00a0Graph logarithmic functions<\/h2>\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 with 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 id=\"Table_04_04_01\" 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>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/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>1<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/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 id=\"Table_04_04_02\" 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\u2019d 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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010821\/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 style=\"text-align: center;\"><strong>Figure 2.\u00a0<\/strong>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<div id=\"fs-id1165137780760\" class=\"note textbox\">\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>= 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<figure id=\"CNX_Precalc_Figure_04_04_003\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010821\/CNX_Precalc_Figure_04_04_003G2.jpg\" alt=\"&quot;Two\" \/><\/figure>\n<p>Figure 3\u00a0shows 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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/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\"><strong>Figure 4.\u00a0<\/strong>The graphs of three logarithmic functions with different bases, all greater than 1.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137871937\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137805513\">How To: Given a logarithmic function with 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>Draw and label the vertical asymptote, <em>x<\/em> = 0.<\/li>\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], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote, <em>x<\/em> = 0.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_03\" class=\"example\">\n<div id=\"fs-id1165137550508\" class=\"exercise\">\n<div id=\"fs-id1165137550510\" class=\"problem textbox shaded\">\n<h3>Example 3: Graphing a Logarithmic Function with the Form\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex].<\/h3>\n<p id=\"fs-id1165137431970\">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=\"q347847\">Show Solution<\/span><\/p>\n<div id=\"q347847\" 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 <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<figure id=\"CNX_Precalc_Figure_04_04_005\" class=\"small\"><span id=\"fs-id1165135508394\"> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/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=\"557\" height=\"419\" \/><\/span><\/figure>\n<p id=\"fs-id1165135697920\" style=\"text-align: center;\"><strong>Figure 5.\u00a0<\/strong>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<\/em> = 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135171582\">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=\"q808887\">Show Solution<\/span><\/p>\n<div id=\"q808887\" 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<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/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.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174289\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174289&theme=oea&iframe_resize_id=ohm174289\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>\u00a0Graphing Transformations of Logarithmic Functions<\/h2>\n<p id=\"fs-id1165137430986\">As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the <strong>parent function<\/strong> [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] without loss of shape.<\/p>\n<section id=\"fs-id1165137734884\">\n<h2>Graphing a Horizontal Shift of\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h2>\n<p>When a constant <em>c<\/em>\u00a0is added to the input of the parent function [latex]f\\left(x\\right)=\\text{log}_{b}\\left(x\\right)[\/latex], the result is a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em>opposite<\/em> direction of the sign on <em>c<\/em>. To visualize horizontal shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] and for <em>c\u00a0<\/em>&gt; 0 alongside the shift left, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], and the shift right, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex].<\/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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/CNX_Precalc_Figure_04_04_007n2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.\" width=\"900\" height=\"526\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165135296307\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Horizontal Shifts of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p id=\"fs-id1165135176174\">For any constant <em>c<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex]<\/p>\n<ul id=\"fs-id1165135206192\">\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\n<li>has domain [latex]\\left(-c,\\infty \\right)[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137641710\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137641715\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], graph the translation.<\/h3>\n<ol id=\"fs-id1165137454284\">\n<li>Identify the horizontal shift:\n<ol id=\"fs-id1165137454288\">\n<li>If <em>c<\/em> &gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>\u00a0units.<\/li>\n<li>If <em>c\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>\u00a0units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting <em>c<\/em>\u00a0from the\u00a0<em>x<\/em>\u00a0coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The Domain is [latex]\\left(-c,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u2013c.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_04\" class=\"example\">\n<div id=\"fs-id1165137414959\" class=\"exercise\">\n<div id=\"fs-id1165137414961\" class=\"problem textbox shaded\">\n<h3>Example 4:\u00a0Graphing a Horizontal Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p id=\"fs-id1165137455420\">Sketch the horizontal shift [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. 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=\"q785817\">Show Solution<\/span><\/p>\n<div id=\"q785817\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137759885\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex], we notice [latex]x+\\left(-2\\right)=x - 2[\/latex].<\/p>\n<p id=\"fs-id1165137784630\">Thus <em>c\u00a0<\/em>= \u20132, so <em>c\u00a0<\/em>&lt; 0. This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] right 2 units.<\/p>\n<p id=\"fs-id1165137836995\">The vertical asymptote is [latex]x=-\\left(-2\\right)[\/latex] or <em>x\u00a0<\/em>= 2.<\/p>\n<p id=\"fs-id1165134042608\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].<\/p>\n<p id=\"fs-id1165137475806\">The new coordinates are found by adding 2 to the <em>x<\/em>\u00a0coordinates.<\/p>\n<p id=\"fs-id1165137748449\">Label the points [latex]\\left(\\frac{7}{3},-1\\right)[\/latex], [latex]\\left(3,0\\right)[\/latex], and [latex]\\left(5,1\\right)[\/latex].<\/p>\n<p>The domain is [latex]\\left(2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.<\/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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010823\/CNX_Precalc_Figure_04_04_0082.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).\" width=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 0.9em;\">\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135329937\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. 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=\"q139882\">Show Solution<\/span><\/p>\n<div id=\"q139882\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-4,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the asymptote <em>x\u00a0<\/em>= \u20134.<span id=\"fs-id1165135209395\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010823\/CNX_Precalc_Figure_04_04_0092.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174300\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174300&theme=oea&iframe_resize_id=ohm174300\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing a Vertical Shift of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/span><\/p>\n<\/section>\n<section id=\"fs-id1165135403538\">When a constant <em>d<\/em>\u00a0is added to the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], the result is a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the direction of the sign on <em>d<\/em>. To visualize vertical shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift up, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] and the shift down, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d[\/latex].<\/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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010823\/CNX_Precalc_Figure_04_04_010F2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.\" width=\"900\" height=\"684\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165137767601\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Vertical Shifts of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p id=\"fs-id1165137661370\">For any constant <em>d<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex]<\/p>\n<ul id=\"fs-id1165137803105\">\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137706002\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137706009\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex], graph the translation.<\/h3>\n<ol>\n<li>Identify the vertical shift:\n<ol>\n<li>If <em>d\u00a0<\/em>&gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>\u00a0units.<\/li>\n<li>If <em>d\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d\u00a0<\/em>units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding <em>d<\/em>\u00a0to the <em>y\u00a0<\/em>coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>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.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_05\" class=\"example\">\n<div id=\"fs-id1165137470057\" class=\"exercise\">\n<div id=\"fs-id1165137470059\" class=\"problem textbox shaded\">\n<h3>Example 5: Graphing a Vertical Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p id=\"fs-id1165137832038\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. 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=\"q657475\">Show Solution<\/span><\/p>\n<div id=\"q657475\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137465913\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex], we will notice <em>d\u00a0<\/em>= \u20132. Thus <em>d\u00a0<\/em>&lt; 0.<\/p>\n<p id=\"fs-id1165135175015\">This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] down 2 units.<\/p>\n<p id=\"fs-id1165137644429\">The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p id=\"fs-id1165137408419\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].<\/p>\n<p id=\"fs-id1165135503945\">The new coordinates are found by subtracting 2 from the <em>y <\/em>coordinates.<\/p>\n<p id=\"fs-id1165135421660\">Label the points [latex]\\left(\\frac{1}{3},-3\\right)[\/latex], [latex]\\left(1,-2\\right)[\/latex], and [latex]\\left(3,-1\\right)[\/latex].<\/p>\n<p id=\"fs-id1165135195524\">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<\/em> = 0.<span id=\"fs-id1165134393856\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010823\/CNX_Precalc_Figure_04_04_0112.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 1).\" \/><\/span><\/p>\n<p id=\"fs-id1165137698285\" style=\"text-align: center;\"><strong>Figure 9.\u00a0<\/strong>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<\/em> = 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137760886\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)+2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. 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=\"q597513\">Show Solution<\/span><\/p>\n<div id=\"q597513\" 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-id1165137874471\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_0122.jpg\" alt=\"Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174304\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174304&theme=oea&iframe_resize_id=ohm174304\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing Stretches and Compressions of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/span><\/p>\n<\/section>\n<section id=\"fs-id1165137770245\">When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by a constant <em>a<\/em> &gt; 0, the result is a <strong>vertical stretch<\/strong> or <strong>compression<\/strong> of the original graph. To visualize stretches and compressions, we set <em>a\u00a0<\/em>&gt; 1 and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the vertical stretch, [latex]g\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] and the vertical compression, [latex]h\\left(x\\right)=\\frac{1}{a}{\\mathrm{log}}_{b}\\left(x\\right)[\/latex].<img decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_013n2.jpg\" alt=\"&quot;Graph\" \/><\/p>\n<div id=\"fs-id1165137433996\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Vertical Stretches and Compressions of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p id=\"fs-id1165137758179\">For any constant <em>a<\/em> &gt; 1, the function [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/p>\n<ul id=\"fs-id1165137428102\">\n<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if <em>a\u00a0<\/em>&gt; 1.<\/li>\n<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if 0 &lt; <em>a\u00a0<\/em>&lt; 1.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>has the <em>x<\/em>-intercept [latex]\\left(1,0\\right)[\/latex].<\/li>\n<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135169301\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135169307\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], [latex]a>0[\/latex], graph the translation.<\/h3>\n<ol id=\"fs-id1165137464127\">\n<li>Identify the vertical stretch or compressions:\n<ol id=\"eip-id1165134081434\">\n<li>If [latex]|a|>1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is stretched by a factor of <em>a<\/em>\u00a0units.<\/li>\n<li>If [latex]|a|<1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is compressed by a factor of <em>a<\/em>\u00a0units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the <em>y<\/em>\u00a0coordinates by <em>a<\/em>.<\/li>\n<li>Label the three points.<\/li>\n<li>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<\/em> = 0.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_06\" class=\"example\">\n<div id=\"fs-id1165135309914\" class=\"exercise\">\n<div id=\"fs-id1165135309916\" class=\"problem textbox shaded\">\n<h3>Example 6: Graphing a Stretch or Compression of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p id=\"fs-id1165137602128\">Sketch a graph of [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. 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=\"q846570\">Show Solution<\/span><\/p>\n<div id=\"q846570\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135210052\">Since the function is [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex], we will notice <em>a\u00a0<\/em>= 2.<\/p>\n<p id=\"fs-id1165135384321\">This means we will stretch the function [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] by a factor of 2.<\/p>\n<p id=\"fs-id1165135481989\">The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p id=\"fs-id1165137757801\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{4},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(4,1\\right)[\/latex].<\/p>\n<p id=\"fs-id1165135570058\">The new coordinates are found by multiplying the <em>y<\/em>\u00a0coordinates by 2.<\/p>\n<p id=\"fs-id1165137837989\">Label the points [latex]\\left(\\frac{1}{4},-2\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(4,\\text{2}\\right)[\/latex].<\/p>\n<p id=\"fs-id1165135543469\">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-id1165134059742\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_0142.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).\" \/><\/span><\/p>\n<p id=\"fs-id1165135566827\" style=\"text-align: center;\"><strong>Figure 11.\u00a0<\/strong>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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135471122\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. 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=\"q645251\">Show Solution<\/span><\/p>\n<div id=\"q645251\" 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-id1165135332505\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_0152.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1\/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_04_04_07\" class=\"example\">\n<div id=\"fs-id1165134267814\" class=\"exercise\">\n<div id=\"fs-id1165134267816\" class=\"problem textbox shaded\">\n<h3>Example 7: Combining a Shift and a Stretch<\/h3>\n<p id=\"fs-id1165137863045\">Sketch a graph of [latex]f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\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=\"q470378\">Show Solution<\/span><\/p>\n<div id=\"q470378\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137935561\">Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5. The vertical asymptote will be shifted to <em>x\u00a0<\/em>= \u20132. The <em>x<\/em>-intercept will be [latex]\\left(-1,0\\right)[\/latex]. The domain will be [latex]\\left(-2,\\infty \\right)[\/latex]. Two points will help give the shape of the graph: [latex]\\left(-1,0\\right)[\/latex] and [latex]\\left(8,5\\right)[\/latex]. We chose <em>x\u00a0<\/em>= 8 as the <em>x<\/em>-coordinate of one point to graph because when <em>x\u00a0<\/em>= 8, <em>x\u00a0<\/em>+ 2 = 10, the base of the common logarithm.<span id=\"fs-id1165135641650\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010824\/CNX_Precalc_Figure_04_04_0162.jpg\" alt=\"Graph of three functions. The parent function is y=log(x), with an asymptote at x=0. The first translation function y=5log(x+2) has an asymptote at x=-2. The second translation function y=log(x+2) has an asymptote at x=-2.\" \/><\/span><\/p>\n<p id=\"fs-id1165137874883\" style=\"text-align: center;\"><strong>Figure 12.\u00a0<\/strong>The domain is [latex]\\left(-2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u20132.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137838697\">Sketch a graph of the function [latex]f\\left(x\\right)=3\\mathrm{log}\\left(x - 2\\right)+1[\/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=\"q793205\">Show Solution<\/span><\/p>\n<div id=\"q793205\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.<\/p>\n<div id=\"fs-id1165137437228\" class=\"solution\">\n<p><span id=\"fs-id1165135177663\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010825\/CNX_Precalc_Figure_04_04_0172.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" \/><\/span><\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174299\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174299&theme=oea&iframe_resize_id=ohm174299\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing Reflections of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/span><\/p>\n<\/section>\n<section id=\"fs-id1165137629003\">\n<p id=\"fs-id1165135169315\">When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a <strong>reflection<\/strong> about the <em>x<\/em>-axis. When the <em>input<\/em> is multiplied by \u20131, the result is a reflection about the <em>y<\/em>-axis. To visualize reflections, we restrict <em>b\u00a0<\/em>&gt; 1, and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex] and the reflection about the <em>y<\/em>-axis, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010825\/CNX_Precalc_Figure_04_04_018n2.jpg\" alt=\"&quot;Graph\" \/><\/p>\n<div id=\"fs-id1165135190744\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Reflections of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p id=\"fs-id1165137722409\">The function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex]<\/p>\n<ul id=\"fs-id1165137832285\">\n<li>reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis.<\/li>\n<li>has domain, [latex]\\left(0,\\infty \\right)[\/latex], range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and vertical asymptote, <em>x\u00a0<\/em>= 0, which are unchanged from the parent function.<\/li>\n<\/ul>\n<p>The function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/p>\n<ul id=\"fs-id1165137734930\">\n<li>reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y<\/em>-axis.<\/li>\n<li>has domain [latex]\\left(-\\infty ,0\\right)[\/latex].<\/li>\n<li>has range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and vertical asymptote, <em>x\u00a0<\/em>= 0, which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137638830\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137638837\">How To: Given a logarithmic function with the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph a translation.<\/h3>\n<table id=\"Table_04_04_08\" class=\"unnumbered\" summary=\"The first column gives the following instructions of graphing a translation of f(x)=-log_b(x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the x-axis; 4. Draw a smooth curve through the points; 5. State the domain, (0, infinity), the range, (-infinity, infinity), and the vertical asymptote x=0. The second column gives the following instructions of graphing a translation of f(x)=log_b(-x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (-1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the y-axis; 4. Draw a smooth curve through the points; 5. State the domain, (-infinity, 0), the range, (-infinity, infinity), and the vertical asymptote x=0.\">\n<thead>\n<tr>\n<th>[latex]\\text{If }f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\n<th>[latex]\\text{If }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1. Draw the vertical asymptote, <em>x\u00a0<\/em>= 0.<\/td>\n<td>1. Draw the vertical asymptote, <em>x\u00a0<\/em>= 0.<\/td>\n<\/tr>\n<tr>\n<td>2. Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\n<td>2. Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis.<\/td>\n<td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y<\/em>-axis.<\/td>\n<\/tr>\n<tr>\n<td>4. Draw a smooth curve through the points.<\/td>\n<td>4. Draw a smooth curve through the points.<\/td>\n<\/tr>\n<tr>\n<td>5. State the domain, [latex]\\left(0,\\infty \\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\n<td>5. State the domain, [latex]\\left(-\\infty ,0\\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"Example_04_04_08\" class=\"example\">\n<div id=\"fs-id1165137697928\" class=\"exercise\">\n<div id=\"fs-id1165137849033\" class=\"problem textbox shaded\">\n<h3>Example 8: Graphing a Reflection of a Logarithmic Function<\/h3>\n<p id=\"fs-id1165137849038\">Sketch a graph of [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. 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=\"q618451\">Show Solution<\/span><\/p>\n<div id=\"q618451\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137836525\">Before graphing [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex], identify the behavior and key points for the graph.<\/p>\n<ul id=\"fs-id1165137769879\">\n<li>Since <em>b\u00a0<\/em>= 10 is greater than one, we know that the parent function is increasing. Since the <em>input<\/em> value is multiplied by \u20131, <em>f<\/em>\u00a0is a reflection of the parent graph about the <em>y-<\/em>axis. Thus, [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] will be decreasing as <em>x<\/em>\u00a0moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>The <em>x<\/em>-intercept is [latex]\\left(-1,0\\right)[\/latex].<\/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<figure id=\"CNX_Precalc_Figure_04_04_019\" class=\"small\"><span id=\"fs-id1165134042188\"> <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010825\/CNX_Precalc_Figure_04_04_0192.jpg\" alt=\"Graph of two functions. The parent function is y=log(x), with an asymptote at x=0 and labeled points at (1, 0), and (10, 0).The translation function f(x)=log(-x) has an asymptote at x=0 and labeled points at (-1, 0) and (-10, 1).\" \/><\/span><\/figure>\n<p id=\"fs-id1165134042202\" style=\"text-align: center;\"><strong>Figure 14.\u00a0<\/strong>The domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135681852\">Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\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=\"q485222\">Show Solution<\/span><\/p>\n<div id=\"q485222\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<span id=\"fs-id1165137855148\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010825\/CNX_Precalc_Figure_04_04_0202.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134579621\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165134579627\">How To: Given a logarithmic equation, use a graphing calculator to approximate solutions.<\/h3>\n<ol id=\"fs-id1165137431118\">\n<li>Press <strong>[Y=]<\/strong>. Enter the given logarithm equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\n<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\n<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select &#8220;intersect&#8221; and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x<\/em>, for the point(s) of intersection.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_09\" class=\"example\">\n<div id=\"fs-id1165135414229\" class=\"exercise\">\n<div id=\"fs-id1165135414231\" class=\"problem textbox shaded\">\n<h3>Example 9: Approximating the Solution of a Logarithmic Equation<\/h3>\n<p id=\"fs-id1165135414236\">Solve [latex]4\\mathrm{ln}\\left(x\\right)+1=-2\\mathrm{ln}\\left(x - 1\\right)[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q513609\">Show Solution<\/span><\/p>\n<div id=\"q513609\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135193434\">Press <strong>[Y=]<\/strong> and enter [latex]4\\mathrm{ln}\\left(x\\right)+1[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter [latex]-2\\mathrm{ln}\\left(x - 1\\right)[\/latex] next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for <em>x<\/em>\u00a0and \u201310 to 10 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to right of <em>x\u00a0<\/em>= 1.<\/p>\n<p id=\"fs-id1165135245763\">For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em>x<\/em>-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) So, to the nearest thousandth, [latex]x\\approx 1.339[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137639531\">Solve [latex]5\\mathrm{log}\\left(x+2\\right)=4-\\mathrm{log}\\left(x\\right)[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q862673\">Show Solution<\/span><\/p>\n<div id=\"q862673\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x\\approx 3.049[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135528930\">\n<h2>Summarizing Translations of the Logarithmic Function<\/h2>\n<p id=\"fs-id1165135528935\">Now that we have worked with each type of translation for the logarithmic function, we can summarize each in the table below\u00a0to arrive at the general equation for translating exponential functions.<\/p>\n<table id=\"Table_04_04_009\" summary=\"Titled,\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Translations of the Parent Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\">Translation<\/th>\n<th style=\"text-align: center;\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift<\/p>\n<ul id=\"fs-id1165137416971\">\n<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\n<li>Vertically <em>d<\/em>\u00a0units up<\/li>\n<\/ul>\n<\/td>\n<td>[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Stretch and Compress<\/p>\n<ul id=\"fs-id1165137427553\">\n<li>Stretch if [latex]|a|>1[\/latex]<\/li>\n<li>Compression if [latex]|a|<1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>x<\/em>-axis<\/td>\n<td>[latex]y=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>y<\/em>-axis<\/td>\n<td>[latex]y={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all translations<\/td>\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137414493\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Translations of Logarithmic Functions<\/h3>\n<p id=\"fs-id1165137414501\">All translations of the parent logarithmic function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], have the form<\/p>\n<div id=\"fs-id1165135408512\" class=\"equation\" style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/div>\n<p id=\"fs-id1165137734655\">where the parent function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right),b>1[\/latex], is<\/p>\n<ul id=\"fs-id1165137531610\">\n<li>shifted vertically up <em>d<\/em>\u00a0units.<\/li>\n<li>shifted horizontally to the left <em>c<\/em>\u00a0units.<\/li>\n<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\n<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\n<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<p id=\"fs-id1165137725084\">For [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex], the graph of the parent function is reflected about the <em>y<\/em>-axis.<\/p>\n<\/div>\n<div id=\"Example_04_04_10\" class=\"example\">\n<div id=\"fs-id1165135296269\" class=\"exercise\">\n<div id=\"fs-id1165135296271\" class=\"problem textbox shaded\">\n<h3>Example 10: Finding the Vertical Asymptote of a Logarithm Graph<\/h3>\n<p id=\"fs-id1165135296276\">What is the vertical asymptote of [latex]f\\left(x\\right)=-2{\\mathrm{log}}_{3}\\left(x+4\\right)+5[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841705\">Show Solution<\/span><\/p>\n<div id=\"q841705\" class=\"hidden-answer\" style=\"display: none\">\n<p>The vertical asymptote is at <em>x\u00a0<\/em>= \u20134.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137871960\">The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to <em>x\u00a0<\/em>= \u20134.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135368433\">What is the vertical asymptote of [latex]f\\left(x\\right)=3+\\mathrm{ln}\\left(x - 1\\right)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q975284\">Show Solution<\/span><\/p>\n<div id=\"q975284\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>x\u00a0<\/em>= 1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_04_04_11\" class=\"example\">\n<div id=\"fs-id1165137849555\" class=\"exercise\">\n<div id=\"fs-id1165137849558\" class=\"problem textbox shaded\">\n<h3>Example 11: Finding the Equation from a Graph<\/h3>\n<p id=\"fs-id1165137849563\">Find a possible equation for the common logarithmic function graphed in Figure 15.<\/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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005323\/CNX_Precalc_Figure_04_04_021.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q993624\">Show Solution<\/span><\/p>\n<div id=\"q993624\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135342979\">This graph has a vertical asymptote at <em>x\u00a0<\/em>= \u20132 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+k[\/latex]<\/p>\n<p id=\"fs-id1165135406913\">It appears the graph passes through the points [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(2,-1\\right)[\/latex]. Substituting [latex]\\left(-1,1\\right)[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&1=-a\\mathrm{log}\\left(-1+2\\right)+k && \\text{Substitute }\\left(-1,1\\right). \\\\ &1=-a\\mathrm{log}\\left(1\\right)+k && \\text{Arithmetic}. \\\\ &1=k && \\text{log(1)}=0. \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137628655\">Next, substituting in [latex]\\left(2,-1\\right)[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&-1=-a\\mathrm{log}\\left(2+2\\right)+1 && \\text{Plug in }\\left(2,-1\\right). \\\\ &-2=-a\\mathrm{log}\\left(4\\right) && \\text{Arithmetic}. \\\\ &a=\\frac{2}{\\mathrm{log}\\left(4\\right)}&& \\text{Solve for }a. \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135192211\">This gives us the equation [latex]f\\left(x\\right)=-\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137735586\">We can verify this answer by comparing the function values in the table below\u00a0with the points on the graph in Example 11.<\/p>\n<table id=\"Table_04_04_010\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>\u22120.58496<\/td>\n<td>\u22121<\/td>\n<td>\u22121.3219<\/td>\n<\/tr>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\n<td>\u22121.5850<\/td>\n<td>\u22121.8074<\/td>\n<td>\u22122<\/td>\n<td>\u22122.1699<\/td>\n<td>\u22122.3219<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137665487\">Give the equation of the natural logarithm graphed in Figure 16.<\/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-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005324\/CNX_Precalc_Figure_04_04_022.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q537017\">Show Solution<\/span><\/p>\n<div id=\"q537017\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x+3\\right)-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137855236\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165137855242\"><strong>Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?<\/strong><\/p>\n<p id=\"fs-id1165137827126\"><em>Yes, if we know the function is a general logarithmic function. For example, look at the graph in Try It 11. The graph approaches x = \u20133 (or thereabouts) more and more closely, so x = \u20133 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex]\\left\\{x|x>-3\\right\\}[\/latex]. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as [latex]x\\to -{3}^{+},f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex].<\/em><\/p>\n<\/div>\n<\/section>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span><\/p>\n<section id=\"fs-id1165137749167\" class=\"key-equations\">\n<table id=\"fs-id1737642\" summary=\"...\">\n<tbody>\n<tr>\n<td>General Form for the Translation of the Parent Logarithmic Function [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137863125\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137863132\">\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 id=\"fs-id1165135441773\">\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<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] horizontally\n<ul id=\"fs-id1165135512562\">\n<li>left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\n<li>right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<\/li>\n<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically\n<ul id=\"fs-id1165137761068\">\n<li>up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\n<li>down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<\/li>\n<li>For any constant <em>a\u00a0<\/em>&gt; 0, the equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\n<ul id=\"fs-id1165134040579\">\n<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if |<em>a<\/em>| &gt; 1.<\/li>\n<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if |<em>a<\/em>| &lt; 1.<\/li>\n<\/ul>\n<\/li>\n<li>When the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a reflection about the <em>x<\/em>-axis. When the input is multiplied by \u20131, the result is a reflection about the <em>y<\/em>-axis.\n<ul id=\"fs-id1165135186594\">\n<li>The equation [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] represents a reflection of the parent function about the <em>x-<\/em>axis.<\/li>\n<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex] represents a reflection of the parent function about the <em>y-<\/em>axis.<\/li>\n<\/ul>\n<ul id=\"fs-id1165137834414\">\n<li>A graphing calculator may be used to approximate solutions to some logarithmic equations.<\/li>\n<\/ul>\n<\/li>\n<li>All translations of the logarithmic function can be summarized by the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex].<\/li>\n<li>Given an equation with the general form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can identify the vertical asymptote <em>x\u00a0<\/em>= \u2013c for the transformation.<\/li>\n<li>Using the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can write the equation of a logarithmic function given its graph.<\/li>\n<\/ul>\n<\/section>\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-100\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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>","protected":false},"author":23485,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-100","chapter","type-chapter","status-publish","hentry"],"part":96,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/100\/revisions"}],"predecessor-version":[{"id":677,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/100\/revisions\/677"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/parts\/96"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/100\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/media?parent=100"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=100"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/contributor?post=100"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/license?post=100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}