{"id":352,"date":"2019-02-08T21:27:09","date_gmt":"2019-02-08T21:27:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-logarithmic-functions\/"},"modified":"2025-03-31T20:30:34","modified_gmt":"2025-03-31T20:30:34","slug":"graphs-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-logarithmic-functions\/","title":{"raw":"2.6 Graphs of Logarithmic Functions","rendered":"2.6 Graphs of Logarithmic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Identify the domain of a logarithmic function and its transformations.<\/li>\r\n \t<li>Graph logarithmic functions and its transformations.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165135194555\">In Section 2.3, <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-exponential-functions\/\">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.<\/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 [latex]t[\/latex] can be found with the equation [latex]A=2500{e}^{0.05t}.[\/latex]<\/p>\r\n<p id=\"fs-id1165137668181\">But what if we wanted to know the year given any balance? We would need to create a corresponding new function by using logarithms to solve for the year; 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? <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_001\">Figure 1<\/a>\u00a0shows this point on the logarithmic graph.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_001\" class=\"wp-caption aligncenter\" style=\"width: 961px;\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"621\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212455\/CNX_Precalc_Figure_04_04_001.jpg\" alt=\"A graph titled, \u201cLogarithmic Model Showing Years as a Function of the Balance in the Account\u201d. The x-axis is labeled, \u201cAccount Balance\u201d, and the y-axis is labeled, \u201cYears\u201d. The line starts at 25,000 on the first year. The graph also notes that the balance reaches 5,000 near year 14.\" width=\"621\" height=\"317\" \/> <strong>Figure 1.<\/strong>[\/caption]\r\n\r\n<\/div>\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<div id=\"fs-id1165137923503\" class=\"bc-section section\">\r\n<h3>Finding the Domain of a Logarithmic Function<\/h3>\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]f\\left(x\\right)={b}^{x}\\text{ }[\/latex] for any real number [latex]x[\/latex] and 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 [latex]f\\left(x\\right)[\/latex] is [latex]\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The range of [latex]f\\left(x\\right)[\/latex] is [latex]\\left(0,\\infty \\right).[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135641666\">In a previous section we learned that the logarithmic function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the inverse of the exponential function with base b.\u00a0 So, as inverse functions:<\/p>\r\n\r\n<ul id=\"fs-id1165137656096\">\r\n \t<li>The domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the range of [latex]f^{-1}\\left(x\\right)={b}^{x}:[\/latex] [latex]\\left(0,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The range of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the domain of [latex]f^{-1}\\left(x\\right)={b}^{x}:[\/latex] [latex]\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135245571\">Transformations of the logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] behave similarly to those of other functions. Just as with toolkit and exponential functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections\u2014to the original function without loss of shape.<\/p>\r\n<p id=\"fs-id1165137653624\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-exponential-functions\/\">Section 2.3, 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 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 we can only take the logarithm<em>\u00a0of positive real numbers<\/em>. That is, the value of the input expression 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}}\\left(2x-3\\right).[\/latex] This function is defined for any values of [latex]x[\/latex] such that the input expression, in this case [latex]2x-3,[\/latex] is greater than zero. To find the domain, we set up an inequality and solve for [latex]x:[\/latex]<\/p>\r\n\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}2x-3&amp;&gt;0&amp;&amp;\\text{Show the input expression is greater than zero}.\\\\2x&amp;&gt;3&amp;&amp;\\text{Add 3}.\\\\x&amp;&gt;1.5&amp;&amp;\\text{Divide by 2.}\\end{align*}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165137645047\">In interval notation, the domain of [latex]f\\left(x\\right)={\\mathrm{log}}\\left(2x-3\\right)[\/latex] is [latex]\\left(1.5,\\infty \\right).[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137423048\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135173951\"><strong>Given a logarithmic function, identify the domain. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137823224\" type=\"1\">\r\n \t<li>Set up an inequality showing the input expression greater than zero.<\/li>\r\n \t<li>Solve for [latex]x.[\/latex]<\/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=\"textbox examples\">\r\n<div id=\"fs-id1165137846475\">\r\n<div id=\"fs-id1165137460694\">\r\n<h3>example 1:\u00a0 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\r\n<\/div>\r\n<div id=\"fs-id1165137894538\">[reveal-answer q=\"fs-id1165137894538\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137894538\"]\r\n<p id=\"fs-id1165137693442\">The logarithmic function is defined only when the input expression is positive, so this function is defined when [latex]x+3&gt;0.[\/latex] Solving this inequality,<\/p>\r\n\r\n<div id=\"eip-id1165135381135\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}x+3&amp;&gt;0&amp;&amp;\\text{The input expression must be positive}.\\\\x&amp;&gt;-3&amp;&amp;\\text{Subtract 3}.\\end{align*}[\/latex]<\/div>\r\n&nbsp;\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 id=\"fs-id1165135692225\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_04_04_01\">\r\n<div id=\"fs-id1165134274544\">\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\r\n<\/div>\r\n<div id=\"fs-id1165137651038\">[reveal-answer q=\"fs-id1165137651038\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137651038\"]\r\n<p id=\"fs-id1165137416167\">[latex]\\left(2,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_04_04_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137894615\">\r\n<div id=\"fs-id1165134108527\">\r\n<h3>example 2:\u00a0 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\r\n<\/div>\r\n<div id=\"fs-id1165137925255\">[reveal-answer q=\"fs-id1165137925255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137925255\"]\r\n<p id=\"fs-id1165137780875\">The logarithmic function is defined only when the input expression is positive, so this function is defined when [latex]5-2x&gt;0.[\/latex] Solving this inequality,<\/p>\r\n\r\n<div id=\"eip-id1165135470032\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}5-2x&amp;&gt;0&amp;&amp;\\text{The input expression must be positive}.\\\\-2x&amp;&gt;-5&amp;&amp;\\text{Subtract }5.\\\\x&amp;&lt;\\frac{5}{2}&amp;&amp;\\text{Divide by }-2\\text{ and switch the inequality}.\\end{align*}[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165137656879\">The domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5-2x\\right)[\/latex] is [latex]\\left(\u2013\\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 id=\"fs-id1165137874699\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"ti_04_04_02\">\r\n<div id=\"fs-id1165135188422\">\r\n\r\nWhat is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(x-5\\right)+2?[\/latex]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135187083\">[reveal-answer q=\"fs-id1165135187083\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135187083\"]\r\n<p id=\"fs-id1165137411510\">[latex]\\left(5,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137554026\" class=\"bc-section section\">\r\n<h3>Graphing Logarithmic Functions<\/h3>\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 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 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 base b, 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 <a class=\"autogenerated-content\" href=\"#Table_04_04_01\">Table 1<\/a><strong>.<\/strong><\/p>\r\n\r\n<table id=\"Table_04_04_01\" summary=\"Three rows and eight columns. The first row is labeled, \u201cx\u201d, the second row is labeled, \u201cy=2^x\u201d, and the third row is labeled, \u201clog_2(y)=x\u201d. Reading the columns as ordered pairs, we have the following values for the second row: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8). For the third row: (1\/8, -3), (1\/4, -2), (1\/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 811.5px;\" colspan=\"8\"><strong>Table 1<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 83.5px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 83.5px;\">[latex]-3[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 83.5px;\">[latex]-2[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 83.5px;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 79.5px;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 79.5px;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 79.5px;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 79.5px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 83.5px;\"><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 83.5px;\">[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 83.5px;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 83.5px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 79.5px;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 79.5px;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 79.5px;\">[latex]4[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 79.5px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"border-collapse: collapse;\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\"><strong>[latex]y={2}^{x}[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]4[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\"><strong>[latex]x={\\mathrm{log}}_{2}\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]-3[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]-2[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135509175\">Using the inputs and outputs from <a class=\"autogenerated-content\" href=\"#Table_04_04_01\">Table 1<\/a>, 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] See <a class=\"autogenerated-content\" href=\"#Table_04_04_02\">Table 2<\/a><strong>.<\/strong><\/p>\r\n\r\n<table id=\"Table_04_04_02\" style=\"height: 66px;\" summary=\"Two rows and eight columns. The first row is labeled, \u201cf(x)=2^x\u201d, with the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8). The second row is labeled, \u201cg(x)=log_2(x)\u201d, with the following values: (1\/8, -3), (1\/4, -2), (1\/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 184.5px; text-align: center;\" colspan=\"8\"><strong>Table 2<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td class=\"border\" style=\"height: 26px; width: 184.5px; text-align: center;\"><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 26px; width: 115.5px; text-align: center;\">[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 26px; width: 115.5px; text-align: center;\">[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 26px; width: 115.5px; text-align: center;\">[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 26px; width: 110.5px; text-align: center;\">[latex]\\left(0,1\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 26px; width: 110.5px; text-align: center;\">[latex]\\left(1,2\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 26px; width: 110.5px; text-align: center;\">[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 26px; width: 110.5px;\">[latex]\\left(3,8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 40px;\">\r\n<td class=\"border\" style=\"height: 40px; width: 184.5px; text-align: center;\"><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 115.5px; text-align: center;\">[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 115.5px; text-align: center;\">[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 115.5px; text-align: center;\">[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 110.5px; text-align: center;\">[latex]\\left(1,0\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 110.5px; text-align: center;\">[latex]\\left(2,1\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 110.5px; text-align: center;\">[latex]\\left(4,2\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 110.5px;\">[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. <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_002\">Figure 2<\/a>\u00a0shows the graph of [latex]f[\/latex] and [latex]g.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_002\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212501\/CNX_Precalc_Figure_04_04_002.jpg\" alt=\"Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.\" width=\"487\" height=\"438\" \/> <strong>Figure 2.\u00a0<\/strong>Notice that the graphs of[latex]\\text{ }f\\left(x\\right)={2}^{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)\\text{ }[\/latex]are reflections about the line[latex]\\text{ }y=x.[\/latex][\/caption]<\/div>\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\nRecall that for exponential growth functions like [latex]f\\left(x\\right)=2^x[\/latex], we observed that as x decreased without bound, [latex]f\\left(x\\right)[\/latex] gets closer and closer to zero from above. We concluded that exponential growth functions of the form [latex]f(x)=ab^x[\/latex] have a horizontal asymptote of [latex]y=0[\/latex] on one side.\u00a0 We created tables supporting this idea numerically.\u00a0 Using the inverse relationship, we will study what happens in the logarithmic function as the input gets closer and closer to zero from the right of zero or through values of x slightly larger than zero.\r\n\r\nTo create a table of values to explore this idea, w<span style=\"font-size: 1rem; text-align: initial;\">e choose input values that get closer and closer to zero from the right side and then evaluate [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right).[\/latex]\u00a0 We choose values slightly larger than zero because the logarithm is defined only for positive values and we want to observe what happens near the boundary of the domain.\u00a0\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">See\u00a0<\/span><a class=\"autogenerated-content\" style=\"font-size: 1rem; text-align: initial;\" href=\"#Table_03_07_003\">Table 3<\/a><span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n<div id=\"fs-id1165137759950\" class=\"bc-section section\">\r\n<table id=\"Table_03_07_003\" summary=\"..\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"height: 14px; width: 232px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 18px; text-align: center;\">0.1<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 25px; text-align: center;\">0.01<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 32px; text-align: center;\">0.001<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 39px; text-align: center;\">0.0001<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td class=\"border\" style=\"height: 14px; width: 232px; text-align: center;\"><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 18px; text-align: center;\">-3.322<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 25px; text-align: center;\">-6.644<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 32px; text-align: center;\">-9.966<\/td>\r\n<td class=\"border\" style=\"height: 14px; width: 39px; text-align: center;\">-13.288<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<span style=\"font-size: 1rem; text-align: initial;\">As [latex]x[\/latex] gets closer and closer to zero from the right (or from the positive side), the function values decrease without bound (go toward minus infinity).\u00a0 Referring back to <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_002\">Figure 2<\/a>,\u00a0 we observe that<\/span>\u00a0the graph of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]\u00a0 decreases without bound or decreases rapidly as [latex]x[\/latex] approaches zero from the right.\r\n\r\nWe use<strong> arrow notation<\/strong>\u00a0to express these ideas. See <a class=\"autogenerated-content\" href=\"#Table_03_07_001\">Table 4<\/a>.\r\n<table id=\"Table_03_07_001\" summary=\"..\"><caption>Table 4:\u00a0 Arrow Notation<\/caption>\r\n<thead>\r\n<tr>\r\n<th class=\"border\">Symbol<\/th>\r\n<th class=\"border\">Meaning<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">[latex]x\\to {a}^{-}[\/latex]<\/td>\r\n<td class=\"border\">[latex]x[\/latex] approaches [latex]a[\/latex] from the left ([latex]x \\lt a[\/latex] but values are increasing to get closer and closer to [latex]a[\/latex])<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]x\\to {a}^{+}[\/latex]<\/td>\r\n<td class=\"border\">[latex]x[\/latex] approaches [latex]a[\/latex] from the right ([latex]x&gt;a[\/latex] but values are decreasing to get closer and closer to [latex]a[\/latex])<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]f\\left(x\\right)\\to \\infty [\/latex]<\/td>\r\n<td class=\"border\">the output goes toward infinity (the output increases without bound)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]f\\left(x\\right)\\to -\\infty [\/latex]<\/td>\r\n<td class=\"border\">the output goes toward negative infinity (the output decreases without bound)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div><\/div>\r\n<div id=\"fs-id1165137759950\" class=\"bc-section section\">[latex]\\\\[\/latex]For the function [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex], we write in arrow notation, as [latex]x\\to {0}^{+},\\text{ }g\\left(x\\right)\\to-\\infty .[\/latex]\u00a0 This behavior demonstrates a <strong>vertical asymptote<\/strong>, which is a vertical line where the graph decreases rapidly\u00a0 as the input values get closer and closer to 0 from the right hand side.[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1165137759950\" class=\"bc-section section\">\r\n<div id=\"fs-id1165137732344\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137561740\">A <strong>vertical asymptote<\/strong> of a graph is a vertical line [latex]x=a[\/latex] where the function's output tends toward positive or negative infinity as the inputs approach [latex]a.[\/latex] We write<\/p>\r\n<p style=\"text-align: center;\">\u00a0as [latex]x\\to a^{-},f\\left(x\\right)\\to \\infty,[\/latex] or as [latex]x\\to a^{+},f\\left(x\\right)\\to \\infty, [\/latex] or<\/p>\r\n<p style=\"text-align: center;\">as [latex]x\\to a^{-},f\\left(x\\right)\\to -\\infty,[\/latex] or as [latex]x\\to a^{+},f\\left(x\\right)\\to -\\infty .[\/latex][latex]\\\\[\/latex]<\/p>\r\nNote that vertical asymptotes may exist as\u00a0[latex]x=a[\/latex] is approached from one side or the other.\u00a0 In the case of the logarithmic function, the domain will only exist on one side of the asymptote so the asymptote will be one-sided.\u00a0 When we study other families of functions, we will see examples where the function increases and\/or decreases rapidly on both sides of the vertical asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nNote that when technology is used to graph logarithmic functions, it often appears that there is a vertical intercept rather than a vertical asymptote. Creating a table of values demonstrates that, in fact, there is a vertical asymptote.<span style=\"font-size: 1em;\">\u00a0<\/span>\r\n<div id=\"fs-id1165137780760\">\r\n<div class=\"textbox shaded\">\r\n<h3>Characteristics of the Graph of the 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 [latex]x[\/latex] and constant [latex]b&gt;0,[\/latex] [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: [latex]x=0[\/latex]<\/li>\r\n \t<li>domain: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\r\n \t<li>range: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/li>\r\n \t<li>horizontal intercept: [latex]\\left(1,0\\right)[\/latex] and key point [latex]\\left(b,1\\right)[\/latex]<\/li>\r\n \t<li>vertical intercept: none<\/li>\r\n \t<li>increasing if [latex]b&gt;1[\/latex] and decreasing if [latex]0 \\lt b \\lt 1[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137464857\">See <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_003\">Figure 3<\/a>.<span id=\"fs-id1165137550075\"><\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"824\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212506\/CNX_Precalc_Figure_04_04_003G.jpg\" alt=\"\" width=\"824\" height=\"367\" \/> <strong>Figure 3.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137871937\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137805513\"><strong>Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] graph the function.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135435529\" type=\"1\">\r\n \t<li>Draw and label the vertical asymptote, [latex]x=0.[\/latex]<\/li>\r\n \t<li>Plot the horizontal 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, [latex]x=0.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137550508\">\r\n<div id=\"fs-id1165137550510\">\r\n<h3>example 3:\u00a0 Graphing a Logarithmic Function with the Form <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>).<\/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\r\n<\/div>\r\n<div id=\"fs-id1165137501967\">[reveal-answer q=\"fs-id1165137501967\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137501967\"]\r\n<p id=\"fs-id1165137501970\">Before graphing, identify the behavior and key points for the graph.<\/p>\r\n\r\n<ul id=\"fs-id1165135497154\">\r\n \t<li>Since [latex]b=5[\/latex] is greater than one, we know the function is increasing. The left end of the graph will approach the vertical asymptote [latex]x=0[\/latex] from the positive side, and the right end behavior will increase slowly without bound.<\/li>\r\n \t<li>The horizontal 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 (see <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_005\">Figure 4<\/a>).<\/li>\r\n<\/ul>\r\n<div id=\"CNX_Precalc_Figure_04_04_005\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"350\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212518\/CNX_Precalc_Figure_04_04_005.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=\"350\" height=\"264\" \/> <strong>Figure 4.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135697920\">The domain is [latex]\\left(0,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and there is a vertical asymptote from the positive side of [latex]x=0.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135161300\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"ti_04_04_03\">\r\n<div id=\"fs-id1165135171580\">\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 vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137407451\">[reveal-answer q=\"fs-id1165137407451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137407451\"]<span id=\"fs-id1165134377926\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212523\/CNX_Precalc_Figure_04_04_006.jpg\" alt=\"Graph of f(x)=log_(1\/5)(x) with labeled points at (1\/5, 1) and (1, 0). The y-axis is the asymptote.\" width=\"320\" height=\"241\" \/><\/span>\r\n<p id=\"fs-id1165135509153\">The domain is [latex]\\left(0,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and there is a vertical asymptote from the positive side of [latex]x=0.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137430980\" class=\"bc-section section\">\r\n<div id=\"fs-id1165137554026\" class=\"bc-section section\">\r\n<div id=\"fs-id1165137780760\">\r\n\r\nFor [latex]b &gt; 1,[\/latex] the base of the logarithm effects how quickly the graph increases as [latex]x[\/latex] gets larger or decreases as [latex]x[\/latex] goes to zero.\u00a0<a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_004\">Figure 5<\/a>\u00a0shows the graphs of three logarithmic functions\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] with the base [latex]b=2,[\/latex]\u00a0[latex]b=e\\approx 2.718,[\/latex] and [latex]b=10[\/latex]. Observe that the graphs compress vertically as the value of the base increases. The key points for the graphs are (2, 1), (e, 1) and (10, 1) respectively showing that base [latex]b=2[\/latex] reaches an output value of 1 more quickly than base e or 10.\r\n<div id=\"CNX_Precalc_Figure_04_04_004\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212510\/CNX_Precalc_Figure_04_04_004.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 5.\u00a0<\/strong>The graphs of three logarithmic functions with different bases, all greater than 1.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>Graphing Transformations of Logarithmic Functions<\/h3>\r\n<p id=\"fs-id1165137430986\">As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of toolkit and exponential functions. We can shift, stretch, compress, and reflect the<span class=\"no-emphasis\">\u00a0function\u00a0<\/span>[latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] without loss of shape.<\/p>\r\n\r\n<div id=\"fs-id1165137734884\" class=\"bc-section section\">\r\n<h4>Graphing a Horizontal Shift of <i>y<\/i>\u00a0= log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\r\n<p id=\"fs-id1165135530294\">We will begin by looking at a horizontal shift of the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right).[\/latex] Consider the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right),[\/latex] where [latex]c[\/latex] is a constant. If [latex]c[\/latex] is positive, then the horizontal shift is to the right and if\u00a0[latex]c[\/latex] is negative, the horizontal shift is to the left. Note that, as we observed in earlier sections, because the general formula for the horizontal shift contains a minus sign, [latex]c[\/latex] will have the opposite sign of what you observe in the formula.<\/p>\r\n\r\n\r\n[caption id=\"attachment_2817\" align=\"aligncenter\" width=\"1017\"]<img class=\"wp-image-2817 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/11155608\/26Fig61.png\" alt=\"Summary of horizontal shifts\" width=\"1017\" height=\"504\" \/> Figure 6[\/caption]\r\n\r\n<div id=\"fs-id1165135296307\">\r\n<div class=\"textbox shaded\">\r\n<h3>Horizontal Shifts of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\r\n<p id=\"fs-id1165135176174\">For any constant [latex]c,[\/latex] 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 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right [latex]c[\/latex] units if [latex]c&gt;0.[\/latex]<\/li>\r\n \t<li>shifts the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left [latex]c[\/latex] units if [latex]c&lt;0.[\/latex]<\/li>\r\n \t<li>has the vertical asymptote [latex]x=c.[\/latex]<\/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>\r\n<div id=\"fs-id1165137641710\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137641715\"><strong>Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right),[\/latex] graph the translation.<\/strong><\/p>\r\n\r\n<ol type=\"1\">\r\n \t<li>Determine the value for [latex]c.[\/latex]\u00a0 It will have the opposite sign of what you see in the simplified formula.<\/li>\r\n \t<li>Identify the horizontal shift:\r\n<ul id=\"fs-id1165137454288\" type=\"a\">\r\n \t<li>If [latex]c&gt;0,[\/latex] shift the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right [latex]c[\/latex] units.<\/li>\r\n \t<li>If [latex]c&lt;0,[\/latex] shift the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left [latex]c[\/latex] units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote [latex]x=c.[\/latex]<\/li>\r\n \t<li>Identify two or three key points from the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right).[\/latex] Find new coordinates for the shifted functions by adding [latex]c[\/latex] to the [latex]x[\/latex] coordinate.<\/li>\r\n \t<li>Label the 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 [latex]x=c.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137414959\">\r\n<div id=\"fs-id1165137414961\">\r\n<h3>example 4:\u00a0 Graphing a Horizontal Shift of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/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 the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right).[\/latex] Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137759883\">[reveal-answer q=\"fs-id1165137759883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137759883\"]\r\n<p id=\"fs-id1165137759885\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x-2\\right),[\/latex] the input expression is [latex]x-\\left(2\\right).[\/latex]\u00a0 Thus [latex]c=2,[\/latex] so [latex]c&gt;0.[\/latex] This means we will shift the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] right 2 units.<\/p>\r\n<p id=\"fs-id1165137836995\">The vertical asymptote is [latex]x=2.[\/latex]<\/p>\r\n<p id=\"fs-id1165134042608\">Consider the two key points from the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right),[\/latex]\u00a0 [latex]\\left(1,0\\right),[\/latex] and [latex]\\left(3,1\\right).[\/latex] The new coordinates are found by adding 2 to the [latex]x[\/latex] coordinates.<\/p>\r\n<p id=\"fs-id1165137748449\">Label the points [latex]\\left(3,0\\right),[\/latex] and [latex]\\left(5,1\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165135349221\">The domain is [latex]\\left(2,\\infty \\right), [\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right), [\/latex] and the vertical asymptote is [latex]x=2.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_008\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"402\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212537\/CNX_Precalc_Figure_04_04_008.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=\"402\" height=\"299\" \/> <strong>Figure 7.<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137831972\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"ti_04_04_04\">\r\n<div id=\"fs-id1165135329935\">\r\n<p id=\"fs-id1165135329937\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex] alongside the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134373485\">[reveal-answer q=\"fs-id1165134373485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134373485\"]<span id=\"fs-id1165135209395\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212542\/CNX_Precalc_Figure_04_04_009.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).\" width=\"369\" height=\"275\" \/><\/span>\r\n<p id=\"fs-id1165134240944\">The domain is [latex]\\left(-4,\\infty \\right),[\/latex] the range [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote [latex]x=\u20134.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135403538\" class=\"bc-section section\">\r\n<h4>Graphing a Vertical Shift of <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\r\n<p id=\"fs-id1165134310784\">When a constant [latex]d[\/latex] is added to the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] the result is a <span class=\"no-emphasis\">vertical shift\u00a0<\/span>[latex]d[\/latex] units in the direction of the sign of [latex]d.[\/latex] To visualize vertical shifts, we can observe the general graph of the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift, [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d.[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_04_04_010\">Figure 8<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"attachment_2849\" align=\"aligncenter\" width=\"994\"]<img class=\"wp-image-2849 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/12141902\/26Fig81.png\" alt=\"\" width=\"994\" height=\"504\" \/> Figure 8[\/caption]\r\n\r\n<div id=\"Figure_04_04_010\" class=\"wp-caption aligncenter\" style=\"width: 953px;\"><\/div>\r\n<div id=\"fs-id1165137767601\">\r\n<div class=\"textbox shaded\">\r\n<h3>Vertical Shifts of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\r\n<p id=\"fs-id1165137661370\">For any constant [latex]d,[\/latex] 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 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up [latex]d[\/latex] units if [latex]d&gt;0.[\/latex]<\/li>\r\n \t<li>shifts the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down [latex]d[\/latex] units if [latex]d&lt;0.[\/latex]<\/li>\r\n \t<li>has the vertical asymptote [latex]x=0.[\/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>\r\n<div id=\"fs-id1165137706002\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137706009\"><strong>Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d,[\/latex] graph the translation.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137734894\" type=\"1\">\r\n \t<li>Identify the vertical shift:\r\n<ul id=\"fs-id1165137558342\">\r\n \t<li>If [latex]d&gt;0,[\/latex] shift the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up [latex]d[\/latex] units.<\/li>\r\n \t<li>If [latex]d&lt;0,[\/latex] shift the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down [latex]d[\/latex] units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote [latex]x=0.[\/latex]<\/li>\r\n \t<li>Identify two or three key points from the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Find new coordinates for the shifted functions by adding [latex]d[\/latex] to the [latex]y[\/latex] coordinate.<\/li>\r\n \t<li>Label the 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 [latex]x=0.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137470059\">\r\n<h3>example 5:\u00a0 Graphing a Vertical Shift of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/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 the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n[reveal-answer q=\"305690\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"305690\"]\r\n<p id=\"fs-id1165137465913\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2,[\/latex] we observe that\u00a0 [latex]d=\u20132.[\/latex] Thus [latex]d&lt;0.[\/latex] This means we will shift the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] down 2 units.<\/p>\r\n<p id=\"fs-id1165137644429\">The vertical asymptote is [latex]x=0.[\/latex]<\/p>\r\n<p id=\"fs-id1165137408419\">Consider the three key points from the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex], [latex]\\left(\\frac{1}{3},-1\\right),[\/latex] [latex]\\left(1,0\\right),[\/latex] and [latex]\\left(3,1\\right).[\/latex] The new coordinates are found by subtracting 2 from the <em>y <\/em>coordinates. Label the points [latex]\\left(\\frac{1}{3},-3\\right),[\/latex] [latex]\\left(1,-2\\right),[\/latex]\u00a0 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 [latex]x=0.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_011\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"332\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212554\/CNX_Precalc_Figure_04_04_011-1.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).\" width=\"332\" height=\"352\" \/> <strong>Figure 9.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137698285\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135176716\" class=\"precalculus tryit\">\r\n<h3>Try it #5<\/h3>\r\n<div id=\"ti_04_04_05\">\r\n<div id=\"fs-id1165135510769\">\r\n\r\nSketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)+2[\/latex] alongside the function [latex]y={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]. Include the key points and vertical asymptote on the graph. State the domain, range, and vertical asymptote.\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137874464\" class=\"small\">\r\n\r\n[reveal-answer q=\"69724\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"69724\"]\r\n\r\n<span id=\"fs-id1165137874471\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212557\/CNX_Precalc_Figure_04_04_012.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).\" width=\"300\" height=\"292\" \/><\/span>\r\n<p id=\"fs-id1165137874474\">The domain is [latex]\\left(0,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/p>\r\n<p id=\"fs-id1165137874474\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137770245\" class=\"bc-section section\">\r\n<h4>Graphing Vertical Stretches and Compressions of <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\r\n<p id=\"fs-id1165137418005\">When the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by a constant [latex]a&gt;0,[\/latex] the result is a <strong><span class=\"no-emphasis\">vertical stretch<\/span><\/strong> or <strong><span class=\"no-emphasis\">compression<\/span><\/strong> of the original graph. To visualize stretches and compressions, we set [latex]a&gt;0[\/latex] and observe the general graph of the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the vertical stretch or compression, [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right).[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_04_04_013\">Figure 10<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"attachment_2850\" align=\"aligncenter\" width=\"994\"]<img class=\"wp-image-2850 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/12142002\/26Fig101.png\" alt=\"\" width=\"994\" height=\"536\" \/> Figure 10[\/caption]\r\n\r\n<div id=\"Figure_04_04_013\" class=\"wp-caption aligncenter\" style=\"width: 924px;\"><\/div>\r\n<div id=\"fs-id1165137433996\">\r\n<div class=\"textbox shaded\">\r\n<h3>Vertical Stretches and Compressions of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\r\n<p id=\"fs-id1165137758179\">For any constant [latex]a&gt;1,[\/latex] 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 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]a&gt;1.[\/latex]<\/li>\r\n \t<li>compresses the\u00a0 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]0 \\lt a \\lt 1.[\/latex]<\/li>\r\n \t<li>has the vertical asymptote [latex]x=0.[\/latex]<\/li>\r\n \t<li>has the horizontal 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>\r\n<div class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135169307\"><strong>Given a logarithmic function with the form\u00a0 [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right),[\/latex] [latex]a&gt;0,[\/latex] graph the translation.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137464127\" type=\"1\">\r\n \t<li>Identify the vertical stretch or compressions:\r\n<ul id=\"eip-id1165134081434\">\r\n \t<li>If [latex]|a|&gt;1,[\/latex] the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is stretched by a factor of [latex]a[\/latex] units.<\/li>\r\n \t<li>If [latex]|a|&lt;1,[\/latex] the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is compressed by a factor of [latex]a[\/latex] units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote [latex]x=0.[\/latex]<\/li>\r\n \t<li>Identify two or three key points from the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Find new coordinates for the stretched or compressed function by multiplying the [latex]y[\/latex] coordinates by [latex]a.[\/latex]<\/li>\r\n \t<li>Label the 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 [latex]x=0.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135309914\">\r\n<div>\r\n<h3>example 6:\u00a0 Graphing a Stretch or Compression of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/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 the function [latex]y={\\mathrm{log}}_{4}\\left(x\\right)[\/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135210050\">[reveal-answer q=\"fs-id1165135210050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135210050\"]\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 [latex]a=2.[\/latex] This means we will stretch the function [latex]y={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] by a factor of 2.<\/p>\r\n<p id=\"fs-id1165135481989\">The vertical asymptote is [latex]x=0.[\/latex]<\/p>\r\n<p id=\"fs-id1165137757801\">Consider the two key points from the function [latex]y={\\mathrm{log}}_{4}\\left(x\\right)[\/latex],\u00a0 [latex]\\left(1,0\\right),[\/latex] and [latex]\\left(4,1\\right).[\/latex] The new coordinates are found by multiplying the [latex]y[\/latex] coordinates by 2. Label the points [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,\\text{ }\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]\u00a0 See <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_014\">Figure 11<\/a><strong>.<\/strong><\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_014\" class=\"small\">\r\n\r\n[caption id=\"attachment_3247\" align=\"aligncenter\" width=\"404\"]<img class=\"wp-image-3247 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/28173830\/26ex6graph.png\" 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 (4,2)).\" width=\"404\" height=\"232\" \/> <strong>Figure 11.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134079655\" class=\"precalculus tryit\">\r\n<h3>Try it #6<\/h3>\r\n<div id=\"ti_04_04_06\">\r\n<div id=\"fs-id1165135471120\">\r\n<p id=\"fs-id1165135471122\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{2}\\text{ }{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside the function [latex]y={\\mathrm{log}}_{4}\\left(x\\right)[\/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137778952\">[reveal-answer q=\"fs-id1165137778952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137778952\"]<img class=\"aligncenter wp-image-3249 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/28174137\/26TI6graph.png\" alt=\"\" width=\"389\" height=\"239\" \/><span style=\"font-size: 1rem; text-align: initial;\">The domain is [latex]\\left(0,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_04_04_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134267814\">\r\n<div id=\"fs-id1165134267816\">\r\n<h3>example 7:\u00a0 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 vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137935559\">[reveal-answer q=\"fs-id1165137935559\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137935559\"]\r\n<p id=\"fs-id1165137935561\">First, we move the graph of\u00a0[latex]y=\\mathrm{log}\\left(x\\right)[\/latex] left 2 units, then stretch the function vertically by a factor of 5, as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_016\">Figure 12<\/a>. The vertical asymptote will be shifted to [latex]x=-2.[\/latex] 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 [latex]x=8[\/latex] as the <em>x<\/em>-coordinate of one point to graph because when [latex]x=8,[\/latex] [latex]x+2=10,[\/latex] the base of the common logarithm.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_016\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"389\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212611\/CNX_Precalc_Figure_04_04_016.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.\" width=\"389\" height=\"352\" \/> <strong>Figure 12.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137874883\">The domain is [latex]\\left(-2,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=-2.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135560730\" class=\"precalculus tryit\">\r\n<h3>Try it #7<\/h3>\r\n<div id=\"ti_04_04_07\">\r\n<div id=\"fs-id1165137838695\">\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 vertical asymptote.\u00a0 Identify at least 2 key points on\u00a0[latex]f\\left(x\\right)=3\\mathrm{log}\\left(x-2\\right)+1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137437228\">[reveal-answer q=\"fs-id1165137437228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137437228\"]<span id=\"fs-id1165135177663\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212614\/CNX_Precalc_Figure_04_04_017.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" width=\"330\" height=\"297\" \/><\/span>\r\n<p id=\"fs-id1165135516923\">The domain is [latex]\\left(2,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=2.[\/latex] Key points may include [latex]\\left(3,\\text{ }1\\right),[\/latex] and\u00a0[latex]\\left(12,\\text{ }4\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137629003\" class=\"bc-section section\">\r\n<h4>Graphing Reflections of <i>y<\/i>\u00a0= log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\r\n<p id=\"fs-id1165135169315\">When the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by [latex]-1[\/latex] we are negating the output so, the result is a <strong><span class=\"no-emphasis\">reflection<\/span><\/strong> about the <em>x<\/em>-axis. When the <em>input<\/em> is multiplied by [latex]-1,[\/latex] the result is a reflection about the <em>y<\/em>-axis. To visualize reflections, we restrict [latex]b&gt;1,[\/latex] and observe the general graph of the function [latex]y={\\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\r\n\r\n[caption id=\"attachment_2851\" align=\"aligncenter\" width=\"990\"]<img class=\"wp-image-2851 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/12142230\/26Fig13.png\" alt=\"\" width=\"990\" height=\"507\" \/> Figure 13[\/caption]\r\n\r\n<div id=\"fs-id1165135190744\">\r\n<div class=\"textbox shaded\">\r\n<h3>Reflections of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\r\nThe function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex]\r\n<ul>\r\n \t<li>reflects the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis, and<\/li>\r\n \t<li>has domain, [latex]\\left(0,\\infty \\right),[\/latex] range, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and vertical asymptote, [latex]x=0,[\/latex] which are unchanged from the original function.<\/li>\r\n<\/ul>\r\n<div><\/div>\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 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] and<\/li>\r\n \t<li>has range, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and vertical asymptote, [latex]x=0,[\/latex] which are unchanged from the original function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137638837\"><strong>Given a logarithmic function with the\u00a0 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] graph a translation.<\/strong><\/p>\r\n\r\n<table id=\"Table_04_04_08\" class=\"unnumbered\" style=\"height: 312px;\" 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 style=\"height: 12px;\">\r\n<th class=\"border\" style=\"height: 12px; width: 364.5px;\">If [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n<th class=\"border\" style=\"height: 12px; width: 368.5px;\">If [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 44px;\">\r\n<td class=\"border\" style=\"height: 44px; width: 364.5px;\">\r\n<ol id=\"fs-id1165135313711\" type=\"1\">\r\n \t<li>Draw the vertical asymptote, [latex]x=0.[\/latex]<\/li>\r\n<\/ol>\r\n<\/td>\r\n<td class=\"border\" style=\"height: 44px; width: 368.5px;\">\r\n<ol id=\"fs-id1165137770301\" type=\"1\">\r\n \t<li>Draw the vertical asymptote, [latex]x=0.[\/latex]<\/li>\r\n<\/ol>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td class=\"border\" style=\"height: 44px; width: 364.5px;\">\r\n<ol id=\"fs-id1165137698305\" start=\"2\" type=\"1\">\r\n \t<li>Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right).[\/latex]<\/li>\r\n<\/ol>\r\n<\/td>\r\n<td class=\"border\" style=\"height: 44px; width: 368.5px;\">\r\n<ol id=\"fs-id1165135301707\" start=\"2\" type=\"1\">\r\n \t<li>Plot the <em>x-<\/em>intercept, [latex]\\left(-1,0\\right).[\/latex]<\/li>\r\n<\/ol>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 84px;\">\r\n<td class=\"border\" style=\"height: 84px; width: 364.5px;\">\r\n<ol start=\"3\" type=\"1\">\r\n \t<li>Reflect the graph of the function [latex]fy={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis. Key point [latex]\\left(b,\\text{ }1\\right)[\/latex] reflects to\u00a0[latex]\\left(b,\\text{ }-1\\right).[\/latex]<\/li>\r\n<\/ol>\r\n<\/td>\r\n<td class=\"border\" style=\"height: 84px; width: 368.5px;\">\r\n<ol start=\"3\" type=\"1\">\r\n \t<li>Reflect the graph of the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y<\/em>-axis.\u00a0Key point [latex]\\left(b,\\text{ }1\\right)[\/latex] reflects to\u00a0[latex]\\left(-b,\\text{ }1\\right).[\/latex]<\/li>\r\n<\/ol>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td class=\"border\" style=\"height: 44px; width: 364.5px;\">\r\n<ol id=\"fs-id1165137737386\" start=\"4\" type=\"1\">\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n<\/ol>\r\n<\/td>\r\n<td class=\"border\" style=\"height: 44px; width: 368.5px;\">\r\n<ol id=\"fs-id1165134240959\" start=\"4\" type=\"1\">\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n<\/ol>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 84px;\">\r\n<td class=\"border\" style=\"height: 84px; width: 364.5px;\">\r\n<ol id=\"fs-id1165137535756\" start=\"5\" type=\"1\">\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 [latex]x=0.[\/latex]<\/li>\r\n<\/ol>\r\n<\/td>\r\n<td class=\"border\" style=\"height: 84px; width: 368.5px;\">\r\n<ol id=\"fs-id1165135560670\" start=\"5\" type=\"1\">\r\n \t<li>State the domain, [latex]\\left(-\\infty ,0\\right),[\/latex] the range, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote [latex]x=0.[\/latex]<\/li>\r\n<\/ol>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"Example_04_04_08\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137697928\">\r\n<div id=\"fs-id1165137849033\">\r\n<h3>example 8:\u00a0 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 the function [latex]y=\\mathrm{log}\\left(x\\right)[\/latex]. Include the key points and vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137836523\">[reveal-answer q=\"fs-id1165137836523\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137836523\"]\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 [latex]b=10[\/latex] is greater than one, we know that the function [latex]y=\\mathrm{log}\\left(x\\right)[\/latex] is increasing. Since the <em>input<\/em> value is multiplied by [latex]-1,[\/latex] [latex]f[\/latex] is a reflection of the graph of\u00a0[latex]y=\\mathrm{log}\\left(x\\right)[\/latex] about the <em>y-<\/em>axis. Thus, [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] will be decreasing as [latex]x[\/latex] moves from negative infinity to zero, and as x approaches zero from the left or through negative values, the graph will decrease without bound.<\/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 vertical asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\r\n<\/ul>\r\n<div id=\"CNX_Precalc_Figure_04_04_019\" class=\"small\"><\/div>\r\n\r\n[caption id=\"attachment_2852\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-2852\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/12142348\/26Fig14Ex8-300x186.png\" alt=\"\" width=\"300\" height=\"186\" \/> Figure 14[\/caption]\r\n\r\n&nbsp;\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 [latex]x=0.[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135650792\" class=\"precalculus tryit\">\r\n<h3>Try it #8<\/h3>\r\n<div id=\"ti_04_04_08\">\r\n<div id=\"fs-id1165135681850\">\r\n<p id=\"fs-id1165135681852\">Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right).[\/latex] State the domain, range, and vertical asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137836698\">[reveal-answer q=\"fs-id1165137836698\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137836698\"]<span id=\"fs-id1165137855148\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212624\/CNX_Precalc_Figure_04_04_020.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" width=\"387\" height=\"229\" \/><\/span>\r\n<p id=\"fs-id1165137855161\">The domain is [latex]\\left(-\\infty ,0\\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]\u00a0 Since there is both a horizontal and vertical reflection, both the x and y coordinates will be negated.\u00a0 Key points [latex]\\left(1,\\text{ } 0\\right)[\/latex] and [latex]\\left(10,\\text{ } 1\\right)[\/latex] on\u00a0[latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex] will transform to [latex]\\left(-1,\\text{ } 0\\right)[\/latex] and [latex]\\left(-10,\\text{ } -1\\right)[\/latex] for the translated function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135528930\" class=\"bc-section section\">\r\n<h4>Summarizing Translations of the Logarithmic Function<\/h4>\r\nNow that we have worked with each type of translation for the logarithmic function, we can summarize each in <a class=\"autogenerated-content\" href=\"#Table_04_04_009\">Table 5<\/a>\u00a0to arrive at the general equation for translating exponential functions.\r\n<table id=\"Table_04_04_009\"><caption>Table 5<\/caption>\r\n<thead>\r\n<tr>\r\n<th style=\"width: 669.5px; text-align: center;\" colspan=\"2\">Translations of the Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th class=\"border\" style=\"width: 354.5px;\">Translation<\/th>\r\n<th class=\"border\" style=\"width: 302.5px;\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 354.5px;\">Shift\r\n<ul id=\"fs-id1165137416971\">\r\n \t<li>Horizontally [latex]c[\/latex] units to the left or right<\/li>\r\n \t<li>Vertically [latex]d[\/latex] units up or down<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 354.5px;\">Stretch and Compress\r\n<ul id=\"fs-id1165137427553\">\r\n \t<li>Vertical stretch if [latex]|a|&gt;1[\/latex]<\/li>\r\n \t<li>Vertical compression if [latex]|a|&lt;1[\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 354.5px;\">Reflect about the <em>x<\/em>-axis<\/td>\r\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 354.5px;\">Reflect about the <em>y<\/em>-axis<\/td>\r\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 354.5px;\">General equation for all translations<\/td>\r\n<td class=\"border\" style=\"width: 302.5px;\">[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<div id=\"fs-id1165137414493\"><\/div>\r\n<div id=\"Example_04_04_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135296269\">\r\n<div id=\"fs-id1165135296271\">\r\n<h3>example 9:\u00a0 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\r\n<\/div>\r\n<div id=\"fs-id1165137572550\">[reveal-answer q=\"fs-id1165137572550\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137572550\"]\r\n<p id=\"fs-id1165137572552\">The vertical asymptote is [latex]x=-4.[\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nThe coefficient, the base, and the upward translation do not affect the vertical asymptote.\u00a0 The shift of the curve 4 units to the left shifts the vertical asymptote to [latex]x=-4.[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137748682\" class=\"precalculus tryit\">\r\n<h3>Try it #9<\/h3>\r\n<div id=\"ti_04_04_10\">\r\n<div id=\"fs-id1165137748692\">\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\r\n<\/div>\r\n<div id=\"fs-id1165135388504\">[reveal-answer q=\"fs-id1165135388504\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135388504\"]\r\n<p id=\"fs-id1165135511477\">[latex]x=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_04_04_11\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137849555\">\r\n<div id=\"fs-id1165137849558\">\r\n<h3>example 10:\u00a0 Finding the Equation from a Graph<\/h3>\r\n<p id=\"fs-id1165137849563\">Find a possible equation for the common logarithmic function graphed in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_021\">Figure 15<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_021\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"280\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212627\/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=\"280\" height=\"211\" \/> <strong>Figure 15.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135342977\">[reveal-answer q=\"fs-id1165135342977\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135342977\"]\r\n<p id=\"fs-id1165135342979\">This graph has a vertical asymptote at [latex]x=-2[\/latex] 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\r\n<div id=\"eip-id1165133361454\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+d.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135406913\">It appears the graph passes through the points [latex]\\left(\u20131,1\\right)[\/latex] and [latex]\\left(2,-1\\right).[\/latex] Substituting [latex]\\left(\u20131,1\\right),[\/latex]<\/p>\r\n&nbsp;\r\n<div id=\"eip-id1165134101923\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}1&amp;=-a\\mathrm{log}\\left(-1+2\\right)+d &amp;&amp; \\text{Substitute }\\left(-1,1\\right).\\hfill \\\\ 1&amp;=-a\\mathrm{log}\\left(1\\right)+d\\hfill &amp;&amp; \\text{Arithmetic}.\\hfill \\\\ 1&amp;=d\\hfill &amp;&amp; \\text{log(1)}=0.\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\nTherefore, we now have\u00a0[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+1.[\/latex] Next, substituting into this equation the point [latex]\\left(2,\u20131\\right)[\/latex] we have,\r\n<div id=\"eip-id1165135431720\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}-1&amp;=-a\\mathrm{log}\\left(2+2\\right)+1&amp;&amp; \\text{Plug in }\\left(2,-1\\right).\\hfill \\\\ -2&amp;=-a\\mathrm{log}\\left(4\\right)&amp;&amp; \\text{Arithmetic}.\\hfill \\\\a&amp;=\\frac{2}{\\mathrm{log}\\left(4\\right)}\\hfill &amp;&amp; \\text{Solve for }a.\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\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<h3>Analysis<\/h3>\r\nWe can verify this answer by comparing the function values in <a class=\"autogenerated-content\" href=\"#Table_04_04_010\">Table 6<\/a>\u00a0with the points on the graph in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_021\">Figure 15<\/a>.\r\n<table id=\"Table_04_04_010\" style=\"height: 44px;\" summary=\"..\"><caption>Table 6<\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 11px;\">\r\n<td class=\"border\" style=\"height: 11px; width: 138.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 48.6563px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td class=\"border\" style=\"height: 11px; width: 138.656px; text-align: center;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 48.6563px; text-align: center;\">\u22120.58496<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121.3219<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td class=\"border\" style=\"height: 11px; width: 138.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 48.6563px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td class=\"border\" style=\"height: 11px; width: 138.656px; text-align: center;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121.5850<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121.8074<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 48.6563px; text-align: center;\">\u22122<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22122.1699<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\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 id=\"eip-id1525883\" class=\"precalculus tryit\">\r\n<h3>Try it #10<\/h3>\r\n<div id=\"ti_04_04_11\">\r\n<div id=\"fs-id1165137665484\">\r\n<p id=\"fs-id1165137665487\">Give the equation of the natural logarithm graphed in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_022\">Figure 16<\/a>.\u00a0 Note that the points [latex]\\left(-2, -1\\right) [\/latex] and [latex]\\left(e-3, 1\\right)[\/latex] are on the graph.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_022\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"294\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212629\/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=\"294\" height=\"267\" \/> <strong>Figure 16.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137436435\">[reveal-answer q=\"fs-id1165137436435\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137436435\"]\r\n<p id=\"fs-id1165137436437\">[latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x+3\\right)-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137855236\" class=\"precalculus qa key-takeaways\">\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 <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_022\">Figure 16<\/a>. The graph approaches [latex]x=-3[\/latex] (or thereabouts) more and more closely, so [latex]x=-3[\/latex] 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\\text{ }|\\text{ }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<\/div>\r\n<div id=\"fs-id1165137735050\" class=\"precalculus media\">\r\n<p id=\"fs-id1165137735056\">Access these online resources for additional instruction and practice with graphing logarithms.<\/p>\r\n\r\n<ul id=\"fs-id1165137735060\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/graphexplog\">Graph an Exponential Function and Logarithmic Function<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/matchexplog\">Match Graphs with Exponential and Logarithmic Functions<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/domainlog\">Find the Domain of Logarithmic Functions<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137749167\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"fs-id1737642\" summary=\"...\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 609.375px;\">General Form for the Translation of the Logarithmic Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 352.625px;\">[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<\/div>\r\n<div id=\"fs-id1165137863125\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165137863132\">\r\n \t<li>To find the domain of a logarithmic function, set up an inequality showing the input expression greater than zero, and solve for [latex]x.[\/latex]<\/li>\r\n \t<li>The graph of the 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 [latex]x=0,[\/latex] and\r\n<ul id=\"fs-id1165135441773\">\r\n \t<li>if [latex]b&gt;1,[\/latex] the function is increasing.<\/li>\r\n \t<li>if [latex]0 \\lt b \\lt1,[\/latex] the function is decreasing.<\/li>\r\n<\/ul>\r\n&nbsp;<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex] shifts the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] horizontally\r\n<ul id=\"fs-id1165135512562\">\r\n \t<li>right [latex]c[\/latex] units if [latex]c&gt;0.[\/latex]<\/li>\r\n \t<li>left [latex]c[\/latex] units if [latex]c \\lt 0.[\/latex]<\/li>\r\n<\/ul>\r\n&nbsp;<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] shifts the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically\r\n<ul>\r\n \t<li>up [latex]d[\/latex] units if [latex]d&gt;0.[\/latex]<\/li>\r\n \t<li>down [latex]d[\/latex] units if [latex]d \\lt 0.[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\r\n<ul>\r\n \t<li>stretches the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]|a|&gt;1.[\/latex]<\/li>\r\n \t<li>compresses the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]|a|&lt;1.[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] represents a reflection of the function\u00a0[latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] 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\u00a0 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y-<\/em>axis.<\/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 [latex]x=c[\/latex] 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<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Identify the domain of a logarithmic function and its transformations.<\/li>\n<li>Graph logarithmic functions and its transformations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135194555\">In Section 2.3, <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-exponential-functions\/\">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.<\/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 [latex]t[\/latex] can be found with the equation [latex]A=2500{e}^{0.05t}.[\/latex]<\/p>\n<p id=\"fs-id1165137668181\">But what if we wanted to know the year given any balance? We would need to create a corresponding new function by using logarithms to solve for the year; 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? <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_001\">Figure 1<\/a>\u00a0shows this point on the logarithmic graph.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_001\" class=\"wp-caption aligncenter\" style=\"width: 961px;\">\n<div style=\"width: 631px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212455\/CNX_Precalc_Figure_04_04_001.jpg\" alt=\"A graph titled, \u201cLogarithmic Model Showing Years as a Function of the Balance in the Account\u201d. The x-axis is labeled, \u201cAccount Balance\u201d, and the y-axis is labeled, \u201cYears\u201d. The line starts at 25,000 on the first year. The graph also notes that the balance reaches 5,000 near year 14.\" width=\"621\" height=\"317\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/p>\n<\/div>\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<div id=\"fs-id1165137923503\" class=\"bc-section section\">\n<h3>Finding the Domain of a Logarithmic Function<\/h3>\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]f\\left(x\\right)={b}^{x}\\text{ }[\/latex] for any real number [latex]x[\/latex] and constant [latex]b>0,[\/latex] [latex]b\\ne 1,[\/latex] where<\/p>\n<ul id=\"fs-id1165137736024\">\n<li>The domain of [latex]f\\left(x\\right)[\/latex] is [latex]\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\n<li>The range of [latex]f\\left(x\\right)[\/latex] is [latex]\\left(0,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135641666\">In a previous section we learned that the logarithmic function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the inverse of the exponential function with base b.\u00a0 So, as inverse functions:<\/p>\n<ul id=\"fs-id1165137656096\">\n<li>The domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the range of [latex]f^{-1}\\left(x\\right)={b}^{x}:[\/latex] [latex]\\left(0,\\infty \\right).[\/latex]<\/li>\n<li>The range of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the domain of [latex]f^{-1}\\left(x\\right)={b}^{x}:[\/latex] [latex]\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135245571\">Transformations of the logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] behave similarly to those of other functions. Just as with toolkit and exponential functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections\u2014to the original function without loss of shape.<\/p>\n<p id=\"fs-id1165137653624\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-exponential-functions\/\">Section 2.3, 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 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 we can only take the logarithm<em>\u00a0of positive real numbers<\/em>. That is, the value of the input expression 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}}\\left(2x-3\\right).[\/latex] This function is defined for any values of [latex]x[\/latex] such that the input expression, in this case [latex]2x-3,[\/latex] is greater than zero. To find the domain, we set up an inequality and solve for [latex]x:[\/latex]<\/p>\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}2x-3&>0&&\\text{Show the input expression is greater than zero}.\\\\2x&>3&&\\text{Add 3}.\\\\x&>1.5&&\\text{Divide by 2.}\\end{align*}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137645047\">In interval notation, the domain of [latex]f\\left(x\\right)={\\mathrm{log}}\\left(2x-3\\right)[\/latex] is [latex]\\left(1.5,\\infty \\right).[\/latex]<\/p>\n<div id=\"fs-id1165137423048\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135173951\"><strong>Given a logarithmic function, identify the domain. <\/strong><\/p>\n<ol id=\"fs-id1165137823224\" type=\"1\">\n<li>Set up an inequality showing the input expression greater than zero.<\/li>\n<li>Solve for [latex]x.[\/latex]<\/li>\n<li>Write the domain in interval notation.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137846475\">\n<div id=\"fs-id1165137460694\">\n<h3>example 1:\u00a0 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>\n<div id=\"fs-id1165137894538\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137894538\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137894538\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137693442\">The logarithmic function is defined only when the input expression is positive, so this function is defined when [latex]x+3>0.[\/latex] Solving this inequality,<\/p>\n<div id=\"eip-id1165135381135\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}x+3&>0&&\\text{The input expression must be positive}.\\\\x&>-3&&\\text{Subtract 3}.\\end{align*}[\/latex]<\/div>\n<p>&nbsp;<\/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 id=\"fs-id1165135692225\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_04_04_01\">\n<div id=\"fs-id1165134274544\">\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>\n<div id=\"fs-id1165137651038\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137651038\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137651038\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137416167\">[latex]\\left(2,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_04_04_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137894615\">\n<div id=\"fs-id1165134108527\">\n<h3>example 2:\u00a0 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>\n<div id=\"fs-id1165137925255\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137925255\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137925255\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137780875\">The logarithmic function is defined only when the input expression is positive, so this function is defined when [latex]5-2x>0.[\/latex] Solving this inequality,<\/p>\n<div id=\"eip-id1165135470032\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}5-2x&>0&&\\text{The input expression must be positive}.\\\\-2x&>-5&&\\text{Subtract }5.\\\\x&<\\frac{5}{2}&&\\text{Divide by }-2\\text{ and switch the inequality}.\\end{align*}[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165137656879\">The domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5-2x\\right)[\/latex] is [latex]\\left(\u2013\\infty ,\\frac{5}{2}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874699\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"ti_04_04_02\">\n<div id=\"fs-id1165135188422\">\n<p>What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(x-5\\right)+2?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135187083\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135187083\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135187083\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137411510\">[latex]\\left(5,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137554026\" class=\"bc-section section\">\n<h3>Graphing Logarithmic Functions<\/h3>\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 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 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 base b, 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 <a class=\"autogenerated-content\" href=\"#Table_04_04_01\">Table 1<\/a><strong>.<\/strong><\/p>\n<table id=\"Table_04_04_01\" summary=\"Three rows and eight columns. The first row is labeled, \u201cx\u201d, the second row is labeled, \u201cy=2^x\u201d, and the third row is labeled, \u201clog_2(y)=x\u201d. Reading the columns as ordered pairs, we have the following values for the second row: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8). For the third row: (1\/8, -3), (1\/4, -2), (1\/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 811.5px;\" colspan=\"8\"><strong>Table 1<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 83.5px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 83.5px;\">[latex]-3[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 83.5px;\">[latex]-2[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 83.5px;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 79.5px;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 79.5px;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 79.5px;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 12px; width: 79.5px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 83.5px;\"><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 83.5px;\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 83.5px;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 83.5px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 79.5px;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 79.5px;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 79.5px;\">[latex]4[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; height: 26px; width: 79.5px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse;\">\n<tbody>\n<tr>\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\"><strong>[latex]y={2}^{x}[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]4[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\"><strong>[latex]x={\\mathrm{log}}_{2}\\left(y\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]-3[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]-2[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 83.5px;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 79.5px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135509175\">Using the inputs and outputs from <a class=\"autogenerated-content\" href=\"#Table_04_04_01\">Table 1<\/a>, 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] See <a class=\"autogenerated-content\" href=\"#Table_04_04_02\">Table 2<\/a><strong>.<\/strong><\/p>\n<table id=\"Table_04_04_02\" style=\"height: 66px;\" summary=\"Two rows and eight columns. The first row is labeled, \u201cf(x)=2^x\u201d, with the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8). The second row is labeled, \u201cg(x)=log_2(x)\u201d, with the following values: (1\/8, -3), (1\/4, -2), (1\/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).\">\n<tbody>\n<tr>\n<td style=\"width: 184.5px; text-align: center;\" colspan=\"8\"><strong>Table 2<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td class=\"border\" style=\"height: 26px; width: 184.5px; text-align: center;\"><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 26px; width: 115.5px; text-align: center;\">[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 26px; width: 115.5px; text-align: center;\">[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 26px; width: 115.5px; text-align: center;\">[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 26px; width: 110.5px; text-align: center;\">[latex]\\left(0,1\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 26px; width: 110.5px; text-align: center;\">[latex]\\left(1,2\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 26px; width: 110.5px; text-align: center;\">[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 26px; width: 110.5px;\">[latex]\\left(3,8\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 40px;\">\n<td class=\"border\" style=\"height: 40px; width: 184.5px; text-align: center;\"><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 40px; width: 115.5px; text-align: center;\">[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 115.5px; text-align: center;\">[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 115.5px; text-align: center;\">[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 110.5px; text-align: center;\">[latex]\\left(1,0\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 110.5px; text-align: center;\">[latex]\\left(2,1\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 110.5px; text-align: center;\">[latex]\\left(4,2\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 110.5px;\">[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. <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_002\">Figure 2<\/a>\u00a0shows the graph of [latex]f[\/latex] and [latex]g.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_002\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212501\/CNX_Precalc_Figure_04_04_002.jpg\" alt=\"Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.\" width=\"487\" height=\"438\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.\u00a0<\/strong>Notice that the graphs of[latex]\\text{ }f\\left(x\\right)={2}^{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)\\text{ }[\/latex]are reflections about the line[latex]\\text{ }y=x.[\/latex]<\/p>\n<\/div>\n<\/div>\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>&#8211; intercept at [latex]\\left(1,0\\right).[\/latex]<\/li>\n<li>The domain of [latex]f\\left(x\\right)={2}^{x},[\/latex] [latex]\\left(-\\infty ,\\infty \\right),[\/latex] is the same as the range of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right).[\/latex]<\/li>\n<li>The range of [latex]f\\left(x\\right)={2}^{x},[\/latex] [latex]\\left(0,\\infty \\right),[\/latex] is the same as the domain of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right).[\/latex]<\/li>\n<\/ul>\n<p>Recall that for exponential growth functions like [latex]f\\left(x\\right)=2^x[\/latex], we observed that as x decreased without bound, [latex]f\\left(x\\right)[\/latex] gets closer and closer to zero from above. We concluded that exponential growth functions of the form [latex]f(x)=ab^x[\/latex] have a horizontal asymptote of [latex]y=0[\/latex] on one side.\u00a0 We created tables supporting this idea numerically.\u00a0 Using the inverse relationship, we will study what happens in the logarithmic function as the input gets closer and closer to zero from the right of zero or through values of x slightly larger than zero.<\/p>\n<p>To create a table of values to explore this idea, w<span style=\"font-size: 1rem; text-align: initial;\">e choose input values that get closer and closer to zero from the right side and then evaluate [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right).[\/latex]\u00a0 We choose values slightly larger than zero because the logarithm is defined only for positive values and we want to observe what happens near the boundary of the domain.\u00a0\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">See\u00a0<\/span><a class=\"autogenerated-content\" style=\"font-size: 1rem; text-align: initial;\" href=\"#Table_03_07_003\">Table 3<\/a><span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<div id=\"fs-id1165137759950\" class=\"bc-section section\">\n<table id=\"Table_03_07_003\" summary=\"..\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"height: 14px; width: 232px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 14px; width: 18px; text-align: center;\">0.1<\/td>\n<td class=\"border\" style=\"height: 14px; width: 25px; text-align: center;\">0.01<\/td>\n<td class=\"border\" style=\"height: 14px; width: 32px; text-align: center;\">0.001<\/td>\n<td class=\"border\" style=\"height: 14px; width: 39px; text-align: center;\">0.0001<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td class=\"border\" style=\"height: 14px; width: 232px; text-align: center;\"><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 14px; width: 18px; text-align: center;\">-3.322<\/td>\n<td class=\"border\" style=\"height: 14px; width: 25px; text-align: center;\">-6.644<\/td>\n<td class=\"border\" style=\"height: 14px; width: 32px; text-align: center;\">-9.966<\/td>\n<td class=\"border\" style=\"height: 14px; width: 39px; text-align: center;\">-13.288<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><span style=\"font-size: 1rem; text-align: initial;\">As [latex]x[\/latex] gets closer and closer to zero from the right (or from the positive side), the function values decrease without bound (go toward minus infinity).\u00a0 Referring back to <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_002\">Figure 2<\/a>,\u00a0 we observe that<\/span>\u00a0the graph of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]\u00a0 decreases without bound or decreases rapidly as [latex]x[\/latex] approaches zero from the right.<\/p>\n<p>We use<strong> arrow notation<\/strong>\u00a0to express these ideas. See <a class=\"autogenerated-content\" href=\"#Table_03_07_001\">Table 4<\/a>.<\/p>\n<table id=\"Table_03_07_001\" summary=\"..\">\n<caption>Table 4:\u00a0 Arrow Notation<\/caption>\n<thead>\n<tr>\n<th class=\"border\">Symbol<\/th>\n<th class=\"border\">Meaning<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\">[latex]x\\to {a}^{-}[\/latex]<\/td>\n<td class=\"border\">[latex]x[\/latex] approaches [latex]a[\/latex] from the left ([latex]x \\lt a[\/latex] but values are increasing to get closer and closer to [latex]a[\/latex])<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]x\\to {a}^{+}[\/latex]<\/td>\n<td class=\"border\">[latex]x[\/latex] approaches [latex]a[\/latex] from the right ([latex]x>a[\/latex] but values are decreasing to get closer and closer to [latex]a[\/latex])<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]f\\left(x\\right)\\to \\infty[\/latex]<\/td>\n<td class=\"border\">the output goes toward infinity (the output increases without bound)<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]f\\left(x\\right)\\to -\\infty[\/latex]<\/td>\n<td class=\"border\">the output goes toward negative infinity (the output decreases without bound)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div><\/div>\n<div id=\"fs-id1165137759950\" class=\"bc-section section\">[latex]\\\\[\/latex]For the function [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex], we write in arrow notation, as [latex]x\\to {0}^{+},\\text{ }g\\left(x\\right)\\to-\\infty .[\/latex]\u00a0 This behavior demonstrates a <strong>vertical asymptote<\/strong>, which is a vertical line where the graph decreases rapidly\u00a0 as the input values get closer and closer to 0 from the right hand side.[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div id=\"fs-id1165137759950\" class=\"bc-section section\">\n<div id=\"fs-id1165137732344\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137561740\">A <strong>vertical asymptote<\/strong> of a graph is a vertical line [latex]x=a[\/latex] where the function&#8217;s output tends toward positive or negative infinity as the inputs approach [latex]a.[\/latex] We write<\/p>\n<p style=\"text-align: center;\">\u00a0as [latex]x\\to a^{-},f\\left(x\\right)\\to \\infty,[\/latex] or as [latex]x\\to a^{+},f\\left(x\\right)\\to \\infty,[\/latex] or<\/p>\n<p style=\"text-align: center;\">as [latex]x\\to a^{-},f\\left(x\\right)\\to -\\infty,[\/latex] or as [latex]x\\to a^{+},f\\left(x\\right)\\to -\\infty .[\/latex][latex]\\\\[\/latex]<\/p>\n<p>Note that vertical asymptotes may exist as\u00a0[latex]x=a[\/latex] is approached from one side or the other.\u00a0 In the case of the logarithmic function, the domain will only exist on one side of the asymptote so the asymptote will be one-sided.\u00a0 When we study other families of functions, we will see examples where the function increases and\/or decreases rapidly on both sides of the vertical asymptote.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Note that when technology is used to graph logarithmic functions, it often appears that there is a vertical intercept rather than a vertical asymptote. Creating a table of values demonstrates that, in fact, there is a vertical asymptote.<span style=\"font-size: 1em;\">\u00a0<\/span><\/p>\n<div id=\"fs-id1165137780760\">\n<div class=\"textbox shaded\">\n<h3>Characteristics of the Graph of the 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 [latex]x[\/latex] and constant [latex]b>0,[\/latex] [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: [latex]x=0[\/latex]<\/li>\n<li>domain: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\n<li>range: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/li>\n<li>horizontal intercept: [latex]\\left(1,0\\right)[\/latex] and key point [latex]\\left(b,1\\right)[\/latex]<\/li>\n<li>vertical intercept: none<\/li>\n<li>increasing if [latex]b>1[\/latex] and decreasing if [latex]0 \\lt b \\lt 1[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137464857\">See <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_003\">Figure 3<\/a>.<span id=\"fs-id1165137550075\"><\/span><\/p>\n<div style=\"width: 834px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212506\/CNX_Precalc_Figure_04_04_003G.jpg\" alt=\"\" width=\"824\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137871937\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137805513\"><strong>Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] graph the function.<\/strong><\/p>\n<ol id=\"fs-id1165135435529\" type=\"1\">\n<li>Draw and label the vertical asymptote, [latex]x=0.[\/latex]<\/li>\n<li>Plot the horizontal 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, [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137550508\">\n<div id=\"fs-id1165137550510\">\n<h3>example 3:\u00a0 Graphing a Logarithmic Function with the Form <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>).<\/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>\n<div id=\"fs-id1165137501967\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137501967\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137501967\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137501970\">Before graphing, identify the behavior and key points for the graph.<\/p>\n<ul id=\"fs-id1165135497154\">\n<li>Since [latex]b=5[\/latex] is greater than one, we know the function is increasing. The left end of the graph will approach the vertical asymptote [latex]x=0[\/latex] from the positive side, and the right end behavior will increase slowly without bound.<\/li>\n<li>The horizontal 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 (see <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_005\">Figure 4<\/a>).<\/li>\n<\/ul>\n<div id=\"CNX_Precalc_Figure_04_04_005\" class=\"small\">\n<div style=\"width: 360px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212518\/CNX_Precalc_Figure_04_04_005.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=\"350\" height=\"264\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135697920\">The domain is [latex]\\left(0,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and there is a vertical asymptote from the positive side of [latex]x=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135161300\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"ti_04_04_03\">\n<div id=\"fs-id1165135171580\">\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 vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137407451\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137407451\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137407451\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165134377926\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212523\/CNX_Precalc_Figure_04_04_006.jpg\" alt=\"Graph of f(x)=log_(1\/5)(x) with labeled points at (1\/5, 1) and (1, 0). The y-axis is the asymptote.\" width=\"320\" height=\"241\" \/><\/span><\/p>\n<p id=\"fs-id1165135509153\">The domain is [latex]\\left(0,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and there is a vertical asymptote from the positive side of [latex]x=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137430980\" class=\"bc-section section\">\n<div id=\"fs-id1165137554026\" class=\"bc-section section\">\n<div id=\"fs-id1165137780760\">\n<p>For [latex]b > 1,[\/latex] the base of the logarithm effects how quickly the graph increases as [latex]x[\/latex] gets larger or decreases as [latex]x[\/latex] goes to zero.\u00a0<a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_004\">Figure 5<\/a>\u00a0shows the graphs of three logarithmic functions\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] with the base [latex]b=2,[\/latex]\u00a0[latex]b=e\\approx 2.718,[\/latex] and [latex]b=10[\/latex]. Observe that the graphs compress vertically as the value of the base increases. The key points for the graphs are (2, 1), (e, 1) and (10, 1) respectively showing that base [latex]b=2[\/latex] reaches an output value of 1 more quickly than base e or 10.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_004\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212510\/CNX_Precalc_Figure_04_04_004.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 5.\u00a0<\/strong>The graphs of three logarithmic functions with different bases, all greater than 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h3>Graphing Transformations of Logarithmic Functions<\/h3>\n<p id=\"fs-id1165137430986\">As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of toolkit and exponential functions. We can shift, stretch, compress, and reflect the<span class=\"no-emphasis\">\u00a0function\u00a0<\/span>[latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] without loss of shape.<\/p>\n<div id=\"fs-id1165137734884\" class=\"bc-section section\">\n<h4>Graphing a Horizontal Shift of <i>y<\/i>\u00a0= log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\n<p id=\"fs-id1165135530294\">We will begin by looking at a horizontal shift of the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right).[\/latex] Consider the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right),[\/latex] where [latex]c[\/latex] is a constant. If [latex]c[\/latex] is positive, then the horizontal shift is to the right and if\u00a0[latex]c[\/latex] is negative, the horizontal shift is to the left. Note that, as we observed in earlier sections, because the general formula for the horizontal shift contains a minus sign, [latex]c[\/latex] will have the opposite sign of what you observe in the formula.<\/p>\n<div id=\"attachment_2817\" style=\"width: 1027px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2817\" class=\"wp-image-2817 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/11155608\/26Fig61.png\" alt=\"Summary of horizontal shifts\" width=\"1017\" height=\"504\" \/><\/p>\n<p id=\"caption-attachment-2817\" class=\"wp-caption-text\">Figure 6<\/p>\n<\/div>\n<div id=\"fs-id1165135296307\">\n<div class=\"textbox shaded\">\n<h3>Horizontal Shifts of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p id=\"fs-id1165135176174\">For any constant [latex]c,[\/latex] 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 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right [latex]c[\/latex] units if [latex]c>0.[\/latex]<\/li>\n<li>shifts the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left [latex]c[\/latex] units if [latex]c<0.[\/latex]<\/li>\n<li>has the vertical asymptote [latex]x=c.[\/latex]<\/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>\n<div id=\"fs-id1165137641710\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137641715\"><strong>Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right),[\/latex] graph the translation.<\/strong><\/p>\n<ol type=\"1\">\n<li>Determine the value for [latex]c.[\/latex]\u00a0 It will have the opposite sign of what you see in the simplified formula.<\/li>\n<li>Identify the horizontal shift:\n<ul id=\"fs-id1165137454288\" type=\"a\">\n<li>If [latex]c>0,[\/latex] shift the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right [latex]c[\/latex] units.<\/li>\n<li>If [latex]c<0,[\/latex] shift the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left [latex]c[\/latex] units.<\/li>\n<\/ul>\n<\/li>\n<li>Draw the vertical asymptote [latex]x=c.[\/latex]<\/li>\n<li>Identify two or three key points from the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right).[\/latex] Find new coordinates for the shifted functions by adding [latex]c[\/latex] to the [latex]x[\/latex] coordinate.<\/li>\n<li>Label the 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 [latex]x=c.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137414959\">\n<div id=\"fs-id1165137414961\">\n<h3>example 4:\u00a0 Graphing a Horizontal Shift of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p id=\"fs-id1165137455420\">Sketch the horizontal shift [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x-2\\right)[\/latex] alongside the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right).[\/latex] Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137759883\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137759883\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137759883\" 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] the input expression is [latex]x-\\left(2\\right).[\/latex]\u00a0 Thus [latex]c=2,[\/latex] so [latex]c>0.[\/latex] This means we will shift the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] right 2 units.<\/p>\n<p id=\"fs-id1165137836995\">The vertical asymptote is [latex]x=2.[\/latex]<\/p>\n<p id=\"fs-id1165134042608\">Consider the two key points from the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right),[\/latex]\u00a0 [latex]\\left(1,0\\right),[\/latex] and [latex]\\left(3,1\\right).[\/latex] The new coordinates are found by adding 2 to the [latex]x[\/latex] coordinates.<\/p>\n<p id=\"fs-id1165137748449\">Label the points [latex]\\left(3,0\\right),[\/latex] and [latex]\\left(5,1\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135349221\">The domain is [latex]\\left(2,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=2.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_008\" class=\"small\">\n<div style=\"width: 412px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212537\/CNX_Precalc_Figure_04_04_008.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=\"402\" height=\"299\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137831972\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"ti_04_04_04\">\n<div id=\"fs-id1165135329935\">\n<p id=\"fs-id1165135329937\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex] alongside the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165134373485\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134373485\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134373485\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165135209395\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212542\/CNX_Precalc_Figure_04_04_009.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).\" width=\"369\" height=\"275\" \/><\/span><\/p>\n<p id=\"fs-id1165134240944\">The domain is [latex]\\left(-4,\\infty \\right),[\/latex] the range [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote [latex]x=\u20134.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135403538\" class=\"bc-section section\">\n<h4>Graphing a Vertical Shift of <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\n<p id=\"fs-id1165134310784\">When a constant [latex]d[\/latex] is added to the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] the result is a <span class=\"no-emphasis\">vertical shift\u00a0<\/span>[latex]d[\/latex] units in the direction of the sign of [latex]d.[\/latex] To visualize vertical shifts, we can observe the general graph of the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift, [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d.[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_04_04_010\">Figure 8<\/a>.<\/p>\n<div id=\"attachment_2849\" style=\"width: 1004px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2849\" class=\"wp-image-2849 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/12141902\/26Fig81.png\" alt=\"\" width=\"994\" height=\"504\" \/><\/p>\n<p id=\"caption-attachment-2849\" class=\"wp-caption-text\">Figure 8<\/p>\n<\/div>\n<div id=\"Figure_04_04_010\" class=\"wp-caption aligncenter\" style=\"width: 953px;\"><\/div>\n<div id=\"fs-id1165137767601\">\n<div class=\"textbox shaded\">\n<h3>Vertical Shifts of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p id=\"fs-id1165137661370\">For any constant [latex]d,[\/latex] 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 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up [latex]d[\/latex] units if [latex]d>0.[\/latex]<\/li>\n<li>shifts the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down [latex]d[\/latex] units if [latex]d<0.[\/latex]<\/li>\n<li>has the vertical asymptote [latex]x=0.[\/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>\n<div id=\"fs-id1165137706002\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137706009\"><strong>Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d,[\/latex] graph the translation.<\/strong><\/p>\n<ol id=\"fs-id1165137734894\" type=\"1\">\n<li>Identify the vertical shift:\n<ul id=\"fs-id1165137558342\">\n<li>If [latex]d>0,[\/latex] shift the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up [latex]d[\/latex] units.<\/li>\n<li>If [latex]d<0,[\/latex] shift the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down [latex]d[\/latex] units.<\/li>\n<\/ul>\n<\/li>\n<li>Draw the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<li>Identify two or three key points from the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Find new coordinates for the shifted functions by adding [latex]d[\/latex] to the [latex]y[\/latex] coordinate.<\/li>\n<li>Label the 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 [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137470059\">\n<h3>example 5:\u00a0 Graphing a Vertical Shift of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p id=\"fs-id1165137832038\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex] alongside the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\n<\/div>\n<div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q305690\">Show Solution<\/span><\/p>\n<div id=\"q305690\" 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 observe that\u00a0 [latex]d=\u20132.[\/latex] Thus [latex]d<0.[\/latex] This means we will shift the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] down 2 units.<\/p>\n<p id=\"fs-id1165137644429\">The vertical asymptote is [latex]x=0.[\/latex]<\/p>\n<p id=\"fs-id1165137408419\">Consider the three key points from the function [latex]y={\\mathrm{log}}_{3}\\left(x\\right)[\/latex], [latex]\\left(\\frac{1}{3},-1\\right),[\/latex] [latex]\\left(1,0\\right),[\/latex] and [latex]\\left(3,1\\right).[\/latex] The new coordinates are found by subtracting 2 from the <em>y <\/em>coordinates. Label the points [latex]\\left(\\frac{1}{3},-3\\right),[\/latex] [latex]\\left(1,-2\\right),[\/latex]\u00a0 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 [latex]x=0.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_011\" class=\"small\">\n<div style=\"width: 342px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212554\/CNX_Precalc_Figure_04_04_011-1.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).\" width=\"332\" height=\"352\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137698285\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176716\" class=\"precalculus tryit\">\n<h3>Try it #5<\/h3>\n<div id=\"ti_04_04_05\">\n<div id=\"fs-id1165135510769\">\n<p>Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)+2[\/latex] alongside the function [latex]y={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]. Include the key points and vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137874464\" class=\"small\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q69724\">Show Solution<\/span><\/p>\n<div id=\"q69724\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1165137874471\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212557\/CNX_Precalc_Figure_04_04_012.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).\" width=\"300\" height=\"292\" \/><\/span><\/p>\n<p id=\"fs-id1165137874474\">The domain is [latex]\\left(0,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/p>\n<p id=\"fs-id1165137874474\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137770245\" class=\"bc-section section\">\n<h4>Graphing Vertical Stretches and Compressions of <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\n<p id=\"fs-id1165137418005\">When the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by a constant [latex]a>0,[\/latex] the result is a <strong><span class=\"no-emphasis\">vertical stretch<\/span><\/strong> or <strong><span class=\"no-emphasis\">compression<\/span><\/strong> of the original graph. To visualize stretches and compressions, we set [latex]a>0[\/latex] and observe the general graph of the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the vertical stretch or compression, [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right).[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_04_04_013\">Figure 10<\/a>.<\/p>\n<div id=\"attachment_2850\" style=\"width: 1004px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2850\" class=\"wp-image-2850 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/12142002\/26Fig101.png\" alt=\"\" width=\"994\" height=\"536\" \/><\/p>\n<p id=\"caption-attachment-2850\" class=\"wp-caption-text\">Figure 10<\/p>\n<\/div>\n<div id=\"Figure_04_04_013\" class=\"wp-caption aligncenter\" style=\"width: 924px;\"><\/div>\n<div id=\"fs-id1165137433996\">\n<div class=\"textbox shaded\">\n<h3>Vertical Stretches and Compressions of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p id=\"fs-id1165137758179\">For any constant [latex]a>1,[\/latex] 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 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]a>1.[\/latex]<\/li>\n<li>compresses the\u00a0 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]0 \\lt a \\lt 1.[\/latex]<\/li>\n<li>has the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<li>has the horizontal 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>\n<div class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135169307\"><strong>Given a logarithmic function with the form\u00a0 [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right),[\/latex] [latex]a>0,[\/latex] graph the translation.<\/strong><\/p>\n<ol id=\"fs-id1165137464127\" type=\"1\">\n<li>Identify the vertical stretch or compressions:\n<ul id=\"eip-id1165134081434\">\n<li>If [latex]|a|>1,[\/latex] the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is stretched by a factor of [latex]a[\/latex] units.<\/li>\n<li>If [latex]|a|<1,[\/latex] the graph of [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is compressed by a factor of [latex]a[\/latex] units.<\/li>\n<\/ul>\n<\/li>\n<li>Draw the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<li>Identify two or three key points from the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Find new coordinates for the stretched or compressed function by multiplying the [latex]y[\/latex] coordinates by [latex]a.[\/latex]<\/li>\n<li>Label the 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 [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135309914\">\n<div>\n<h3>example 6:\u00a0 Graphing a Stretch or Compression of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p id=\"fs-id1165137602128\">Sketch a graph of [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside the function [latex]y={\\mathrm{log}}_{4}\\left(x\\right)[\/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165135210050\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135210050\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135210050\" 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 [latex]a=2.[\/latex] This means we will stretch the function [latex]y={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] by a factor of 2.<\/p>\n<p id=\"fs-id1165135481989\">The vertical asymptote is [latex]x=0.[\/latex]<\/p>\n<p id=\"fs-id1165137757801\">Consider the two key points from the function [latex]y={\\mathrm{log}}_{4}\\left(x\\right)[\/latex],\u00a0 [latex]\\left(1,0\\right),[\/latex] and [latex]\\left(4,1\\right).[\/latex] The new coordinates are found by multiplying the [latex]y[\/latex] coordinates by 2. Label the points [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,\\text{ }\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]\u00a0 See <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_014\">Figure 11<\/a><strong>.<\/strong><\/p>\n<div id=\"CNX_Precalc_Figure_04_04_014\" class=\"small\">\n<div id=\"attachment_3247\" style=\"width: 414px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3247\" class=\"wp-image-3247 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/28173830\/26ex6graph.png\" 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 (4,2)).\" width=\"404\" height=\"232\" \/><\/p>\n<p id=\"caption-attachment-3247\" class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134079655\" class=\"precalculus tryit\">\n<h3>Try it #6<\/h3>\n<div id=\"ti_04_04_06\">\n<div id=\"fs-id1165135471120\">\n<p id=\"fs-id1165135471122\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{2}\\text{ }{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside the function [latex]y={\\mathrm{log}}_{4}\\left(x\\right)[\/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137778952\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137778952\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137778952\" class=\"hidden-answer\" style=\"display: none\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3249 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/28174137\/26TI6graph.png\" alt=\"\" width=\"389\" height=\"239\" \/><span style=\"font-size: 1rem; text-align: initial;\">The domain is [latex]\\left(0,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_04_04_07\" class=\"textbox examples\">\n<div id=\"fs-id1165134267814\">\n<div id=\"fs-id1165134267816\">\n<h3>example 7:\u00a0 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 vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137935559\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137935559\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137935559\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137935561\">First, we move the graph of\u00a0[latex]y=\\mathrm{log}\\left(x\\right)[\/latex] left 2 units, then stretch the function vertically by a factor of 5, as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_016\">Figure 12<\/a>. The vertical asymptote will be shifted to [latex]x=-2.[\/latex] 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 [latex]x=8[\/latex] as the <em>x<\/em>-coordinate of one point to graph because when [latex]x=8,[\/latex] [latex]x+2=10,[\/latex] the base of the common logarithm.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_016\" class=\"small\">\n<div style=\"width: 399px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212611\/CNX_Precalc_Figure_04_04_016.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.\" width=\"389\" height=\"352\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137874883\">The domain is [latex]\\left(-2,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=-2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560730\" class=\"precalculus tryit\">\n<h3>Try it #7<\/h3>\n<div id=\"ti_04_04_07\">\n<div id=\"fs-id1165137838695\">\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 vertical asymptote.\u00a0 Identify at least 2 key points on\u00a0[latex]f\\left(x\\right)=3\\mathrm{log}\\left(x-2\\right)+1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137437228\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137437228\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137437228\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165135177663\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212614\/CNX_Precalc_Figure_04_04_017.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" width=\"330\" height=\"297\" \/><\/span><\/p>\n<p id=\"fs-id1165135516923\">The domain is [latex]\\left(2,\\infty \\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=2.[\/latex] Key points may include [latex]\\left(3,\\text{ }1\\right),[\/latex] and\u00a0[latex]\\left(12,\\text{ }4\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137629003\" class=\"bc-section section\">\n<h4>Graphing Reflections of <i>y<\/i>\u00a0= log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\n<p id=\"fs-id1165135169315\">When the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by [latex]-1[\/latex] we are negating the output so, the result is a <strong><span class=\"no-emphasis\">reflection<\/span><\/strong> about the <em>x<\/em>-axis. When the <em>input<\/em> is multiplied by [latex]-1,[\/latex] the result is a reflection about the <em>y<\/em>-axis. To visualize reflections, we restrict [latex]b>1,[\/latex] and observe the general graph of the function [latex]y={\\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<div id=\"attachment_2851\" style=\"width: 1000px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2851\" class=\"wp-image-2851 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/12142230\/26Fig13.png\" alt=\"\" width=\"990\" height=\"507\" \/><\/p>\n<p id=\"caption-attachment-2851\" class=\"wp-caption-text\">Figure 13<\/p>\n<\/div>\n<div id=\"fs-id1165135190744\">\n<div class=\"textbox shaded\">\n<h3>Reflections of the Function <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p>The function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex]<\/p>\n<ul>\n<li>reflects the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis, and<\/li>\n<li>has domain, [latex]\\left(0,\\infty \\right),[\/latex] range, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and vertical asymptote, [latex]x=0,[\/latex] which are unchanged from the original function.<\/li>\n<\/ul>\n<div><\/div>\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 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] and<\/li>\n<li>has range, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and vertical asymptote, [latex]x=0,[\/latex] which are unchanged from the original function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137638837\"><strong>Given a logarithmic function with the\u00a0 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] graph a translation.<\/strong><\/p>\n<table id=\"Table_04_04_08\" class=\"unnumbered\" style=\"height: 312px;\" 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 style=\"height: 12px;\">\n<th class=\"border\" style=\"height: 12px; width: 364.5px;\">If [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\n<th class=\"border\" style=\"height: 12px; width: 368.5px;\">If [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 44px;\">\n<td class=\"border\" style=\"height: 44px; width: 364.5px;\">\n<ol id=\"fs-id1165135313711\" type=\"1\">\n<li>Draw the vertical asymptote, [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/td>\n<td class=\"border\" style=\"height: 44px; width: 368.5px;\">\n<ol id=\"fs-id1165137770301\" type=\"1\">\n<li>Draw the vertical asymptote, [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td class=\"border\" style=\"height: 44px; width: 364.5px;\">\n<ol id=\"fs-id1165137698305\" start=\"2\" type=\"1\">\n<li>Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right).[\/latex]<\/li>\n<\/ol>\n<\/td>\n<td class=\"border\" style=\"height: 44px; width: 368.5px;\">\n<ol id=\"fs-id1165135301707\" start=\"2\" type=\"1\">\n<li>Plot the <em>x-<\/em>intercept, [latex]\\left(-1,0\\right).[\/latex]<\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<tr style=\"height: 84px;\">\n<td class=\"border\" style=\"height: 84px; width: 364.5px;\">\n<ol start=\"3\" type=\"1\">\n<li>Reflect the graph of the function [latex]fy={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis. Key point [latex]\\left(b,\\text{ }1\\right)[\/latex] reflects to\u00a0[latex]\\left(b,\\text{ }-1\\right).[\/latex]<\/li>\n<\/ol>\n<\/td>\n<td class=\"border\" style=\"height: 84px; width: 368.5px;\">\n<ol start=\"3\" type=\"1\">\n<li>Reflect the graph of the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y<\/em>-axis.\u00a0Key point [latex]\\left(b,\\text{ }1\\right)[\/latex] reflects to\u00a0[latex]\\left(-b,\\text{ }1\\right).[\/latex]<\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td class=\"border\" style=\"height: 44px; width: 364.5px;\">\n<ol id=\"fs-id1165137737386\" start=\"4\" type=\"1\">\n<li>Draw a smooth curve through the points.<\/li>\n<\/ol>\n<\/td>\n<td class=\"border\" style=\"height: 44px; width: 368.5px;\">\n<ol id=\"fs-id1165134240959\" start=\"4\" type=\"1\">\n<li>Draw a smooth curve through the points.<\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<tr style=\"height: 84px;\">\n<td class=\"border\" style=\"height: 84px; width: 364.5px;\">\n<ol id=\"fs-id1165137535756\" start=\"5\" type=\"1\">\n<li>State the domain, [latex]\\left(0,\\infty \\right),[\/latex] the range, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/td>\n<td class=\"border\" style=\"height: 84px; width: 368.5px;\">\n<ol id=\"fs-id1165135560670\" start=\"5\" type=\"1\">\n<li>State the domain, [latex]\\left(-\\infty ,0\\right),[\/latex] the range, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"Example_04_04_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137697928\">\n<div id=\"fs-id1165137849033\">\n<h3>example 8:\u00a0 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 the function [latex]y=\\mathrm{log}\\left(x\\right)[\/latex]. Include the key points and vertical asymptote on the graph. State the domain, range, and vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137836523\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137836523\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137836523\" 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 [latex]b=10[\/latex] is greater than one, we know that the function [latex]y=\\mathrm{log}\\left(x\\right)[\/latex] is increasing. Since the <em>input<\/em> value is multiplied by [latex]-1,[\/latex] [latex]f[\/latex] is a reflection of the graph of\u00a0[latex]y=\\mathrm{log}\\left(x\\right)[\/latex] about the <em>y-<\/em>axis. Thus, [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] will be decreasing as [latex]x[\/latex] moves from negative infinity to zero, and as x approaches zero from the left or through negative values, the graph will decrease without bound.<\/li>\n<li>The <em>x<\/em>-intercept is [latex]\\left(-1,0\\right).[\/latex]<\/li>\n<li>We draw and label the vertical asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\n<\/ul>\n<div id=\"CNX_Precalc_Figure_04_04_019\" class=\"small\"><\/div>\n<div id=\"attachment_2852\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2852\" class=\"size-medium wp-image-2852\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/12142348\/26Fig14Ex8-300x186.png\" alt=\"\" width=\"300\" height=\"186\" \/><\/p>\n<p id=\"caption-attachment-2852\" class=\"wp-caption-text\">Figure 14<\/p>\n<\/div>\n<p>&nbsp;<\/p>\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 [latex]x=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135650792\" class=\"precalculus tryit\">\n<h3>Try it #8<\/h3>\n<div id=\"ti_04_04_08\">\n<div id=\"fs-id1165135681850\">\n<p id=\"fs-id1165135681852\">Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right).[\/latex] State the domain, range, and vertical asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137836698\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137836698\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137836698\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165137855148\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212624\/CNX_Precalc_Figure_04_04_020.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" width=\"387\" height=\"229\" \/><\/span><\/p>\n<p id=\"fs-id1165137855161\">The domain is [latex]\\left(-\\infty ,0\\right),[\/latex] the range is [latex]\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]\u00a0 Since there is both a horizontal and vertical reflection, both the x and y coordinates will be negated.\u00a0 Key points [latex]\\left(1,\\text{ } 0\\right)[\/latex] and [latex]\\left(10,\\text{ } 1\\right)[\/latex] on\u00a0[latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex] will transform to [latex]\\left(-1,\\text{ } 0\\right)[\/latex] and [latex]\\left(-10,\\text{ } -1\\right)[\/latex] for the translated function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135528930\" class=\"bc-section section\">\n<h4>Summarizing Translations of the Logarithmic Function<\/h4>\n<p>Now that we have worked with each type of translation for the logarithmic function, we can summarize each in <a class=\"autogenerated-content\" href=\"#Table_04_04_009\">Table 5<\/a>\u00a0to arrive at the general equation for translating exponential functions.<\/p>\n<table id=\"Table_04_04_009\">\n<caption>Table 5<\/caption>\n<thead>\n<tr>\n<th style=\"width: 669.5px; text-align: center;\" colspan=\"2\">Translations of the Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<tr>\n<th class=\"border\" style=\"width: 354.5px;\">Translation<\/th>\n<th class=\"border\" style=\"width: 302.5px;\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 354.5px;\">Shift<\/p>\n<ul id=\"fs-id1165137416971\">\n<li>Horizontally [latex]c[\/latex] units to the left or right<\/li>\n<li>Vertically [latex]d[\/latex] units up or down<\/li>\n<\/ul>\n<\/td>\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 354.5px;\">Stretch and Compress<\/p>\n<ul id=\"fs-id1165137427553\">\n<li>Vertical stretch if [latex]|a|>1[\/latex]<\/li>\n<li>Vertical compression if [latex]|a|<1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 354.5px;\">Reflect about the <em>x<\/em>-axis<\/td>\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 354.5px;\">Reflect about the <em>y<\/em>-axis<\/td>\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 354.5px;\">General equation for all translations<\/td>\n<td class=\"border\" style=\"width: 302.5px;\">[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x-c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137414493\"><\/div>\n<div id=\"Example_04_04_10\" class=\"textbox examples\">\n<div id=\"fs-id1165135296269\">\n<div id=\"fs-id1165135296271\">\n<h3>example 9:\u00a0 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>\n<div id=\"fs-id1165137572550\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137572550\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137572550\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137572552\">The vertical asymptote is [latex]x=-4.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>The coefficient, the base, and the upward translation do not affect the vertical asymptote.\u00a0 The shift of the curve 4 units to the left shifts the vertical asymptote to [latex]x=-4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137748682\" class=\"precalculus tryit\">\n<h3>Try it #9<\/h3>\n<div id=\"ti_04_04_10\">\n<div id=\"fs-id1165137748692\">\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>\n<div id=\"fs-id1165135388504\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135388504\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135388504\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135511477\">[latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_04_04_11\" class=\"textbox examples\">\n<div id=\"fs-id1165137849555\">\n<div id=\"fs-id1165137849558\">\n<h3>example 10:\u00a0 Finding the Equation from a Graph<\/h3>\n<p id=\"fs-id1165137849563\">Find a possible equation for the common logarithmic function graphed in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_021\">Figure 15<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_021\" class=\"small\">\n<div style=\"width: 290px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212627\/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=\"280\" height=\"211\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 15.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135342977\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135342977\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135342977\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135342979\">This graph has a vertical asymptote at [latex]x=-2[\/latex] 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<div id=\"eip-id1165133361454\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+d.[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135406913\">It appears the graph passes through the points [latex]\\left(\u20131,1\\right)[\/latex] and [latex]\\left(2,-1\\right).[\/latex] Substituting [latex]\\left(\u20131,1\\right),[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div id=\"eip-id1165134101923\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}1&=-a\\mathrm{log}\\left(-1+2\\right)+d && \\text{Substitute }\\left(-1,1\\right).\\hfill \\\\ 1&=-a\\mathrm{log}\\left(1\\right)+d\\hfill && \\text{Arithmetic}.\\hfill \\\\ 1&=d\\hfill && \\text{log(1)}=0.\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p>Therefore, we now have\u00a0[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+1.[\/latex] Next, substituting into this equation the point [latex]\\left(2,\u20131\\right)[\/latex] we have,<\/p>\n<div id=\"eip-id1165135431720\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}-1&=-a\\mathrm{log}\\left(2+2\\right)+1&& \\text{Plug in }\\left(2,-1\\right).\\hfill \\\\ -2&=-a\\mathrm{log}\\left(4\\right)&& \\text{Arithmetic}.\\hfill \\\\a&=\\frac{2}{\\mathrm{log}\\left(4\\right)}\\hfill && \\text{Solve for }a.\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\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<h3>Analysis<\/h3>\n<p>We can verify this answer by comparing the function values in <a class=\"autogenerated-content\" href=\"#Table_04_04_010\">Table 6<\/a>\u00a0with the points on the graph in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_021\">Figure 15<\/a>.<\/p>\n<table id=\"Table_04_04_010\" style=\"height: 44px;\" summary=\"..\">\n<caption>Table 6<\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 11px;\">\n<td class=\"border\" style=\"height: 11px; width: 138.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">0<\/td>\n<td class=\"border\" style=\"height: 11px; width: 48.6563px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">3<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td class=\"border\" style=\"height: 11px; width: 138.656px; text-align: center;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">0<\/td>\n<td class=\"border\" style=\"height: 11px; width: 48.6563px; text-align: center;\">\u22120.58496<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121.3219<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td class=\"border\" style=\"height: 11px; width: 138.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">4<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">5<\/td>\n<td class=\"border\" style=\"height: 11px; width: 48.6563px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">7<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">8<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td class=\"border\" style=\"height: 11px; width: 138.656px; text-align: center;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121.5850<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22121.8074<\/td>\n<td class=\"border\" style=\"height: 11px; width: 48.6563px; text-align: center;\">\u22122<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22122.1699<\/td>\n<td class=\"border\" style=\"height: 11px; width: 42.6563px; text-align: center;\">\u22122.3219<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1525883\" class=\"precalculus tryit\">\n<h3>Try it #10<\/h3>\n<div id=\"ti_04_04_11\">\n<div id=\"fs-id1165137665484\">\n<p id=\"fs-id1165137665487\">Give the equation of the natural logarithm graphed in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_022\">Figure 16<\/a>.\u00a0 Note that the points [latex]\\left(-2, -1\\right)[\/latex] and [latex]\\left(e-3, 1\\right)[\/latex] are on the graph.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_022\" class=\"small\">\n<div style=\"width: 304px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08212629\/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=\"294\" height=\"267\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 16.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436435\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137436435\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137436435\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137436437\">[latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x+3\\right)-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137855236\" class=\"precalculus qa key-takeaways\">\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 <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_022\">Figure 16<\/a>. The graph approaches [latex]x=-3[\/latex] (or thereabouts) more and more closely, so [latex]x=-3[\/latex] 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\\text{ }|\\text{ }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<\/div>\n<div id=\"fs-id1165137735050\" class=\"precalculus media\">\n<p id=\"fs-id1165137735056\">Access these online resources for additional instruction and practice with graphing logarithms.<\/p>\n<ul id=\"fs-id1165137735060\">\n<li><a href=\"http:\/\/openstax.org\/l\/graphexplog\">Graph an Exponential Function and Logarithmic Function<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/matchexplog\">Match Graphs with Exponential and Logarithmic Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/domainlog\">Find the Domain of Logarithmic Functions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137749167\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id1737642\" summary=\"...\">\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 609.375px;\">General Form for the Translation of the Logarithmic Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"width: 352.625px;\">[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x-c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137863125\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137863132\">\n<li>To find the domain of a logarithmic function, set up an inequality showing the input expression greater than zero, and solve for [latex]x.[\/latex]<\/li>\n<li>The graph of the 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 [latex]x=0,[\/latex] and\n<ul id=\"fs-id1165135441773\">\n<li>if [latex]b>1,[\/latex] the function is increasing.<\/li>\n<li>if [latex]0 \\lt b \\lt1,[\/latex] the function is decreasing.<\/li>\n<\/ul>\n<p>&nbsp;<\/li>\n<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex] shifts the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] horizontally\n<ul id=\"fs-id1165135512562\">\n<li>right [latex]c[\/latex] units if [latex]c>0.[\/latex]<\/li>\n<li>left [latex]c[\/latex] units if [latex]c \\lt 0.[\/latex]<\/li>\n<\/ul>\n<p>&nbsp;<\/li>\n<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] shifts the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically\n<ul>\n<li>up [latex]d[\/latex] units if [latex]d>0.[\/latex]<\/li>\n<li>down [latex]d[\/latex] units if [latex]d \\lt 0.[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\n<ul>\n<li>stretches the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]|a|>1.[\/latex]<\/li>\n<li>compresses the function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]|a|<1.[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The equation [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] represents a reflection of the function\u00a0[latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] 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\u00a0 function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y-<\/em>axis.<\/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 [latex]x=c[\/latex] 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<\/div>\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-352\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Graphs of Logarithmic Functions. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:REGENe1G@17\/Graphs-of-Logarithmic-Functions\">https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:REGENe1G@17\/Graphs-of-Logarithmic-Functions<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 1. <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":311,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Graphs of Logarithmic Functions\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:REGENe1G@17\/Graphs-of-Logarithmic-Functions\",\"project\":\"Essential Precalcus, Part 1\",\"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-352","chapter","type-chapter","status-publish","hentry"],"part":223,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/352","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":47,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/352\/revisions"}],"predecessor-version":[{"id":3285,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/352\/revisions\/3285"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/223"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/352\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=352"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=352"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=352"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}