{"id":2455,"date":"2018-07-25T18:14:35","date_gmt":"2018-07-25T18:14:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/?post_type=chapter&#038;p=2455"},"modified":"2018-07-30T20:37:30","modified_gmt":"2018-07-30T20:37:30","slug":"graphs-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/graphs-of-logarithmic-functions\/","title":{"raw":"Graphs of Logarithmic Functions","rendered":"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.<\/li>\r\n \t<li>Graph logarithmic functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165135194555\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/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. In other words, logarithms give the <em>cause<\/em> for an <em>effect<\/em>.<\/p>\r\n<p id=\"fs-id1165137603580\">To illustrate, suppose we invest[latex]\\,\\text{\\$}2500\\,[\/latex]in an account that offers an annual interest rate of[latex]\\,5%,[\/latex]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 for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_001\">(Figure)<\/a> shows this point on the logarithmic graph.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_001\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140536\/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 \ud83d\udcb225,000 on the first year. The graph also notes that the balance reaches \ud83d\udcb25,000 near year 14.\" width=\"975\" height=\"497\" \/> <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]\\,y={b}^{x}\\,[\/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]\\,y\\,[\/latex]is[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The range of[latex]\\,y\\,[\/latex]is[latex]\\,\\left(0,\\infty \\right).[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135641666\">In the last section we learned that the logarithmic function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is the inverse of the exponential function[latex]\\,y={b}^{x}.\\,[\/latex]So, as inverse functions:<\/p>\r\n\r\n<ul id=\"fs-id1165137656096\">\r\n \t<li>The domain of[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is the range of[latex]\\,y={b}^{x}:\\,[\/latex][latex]\\left(0,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The range of[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is the domain of[latex]\\,y={b}^{x}:\\,[\/latex][latex]\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135245571\">Transformations of the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections\u2014to the parent function without loss of shape.<\/p>\r\n<p id=\"fs-id1165137653624\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/graphs-of-exponential-functions\/\">Graphs of Exponential Functions<\/a> we saw that certain transformations can change the <em>range<\/em> of[latex]\\,y={b}^{x}.\\,[\/latex]Similarly, applying transformations to the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]can change the <em>domain<\/em>. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists <em>only of positive real numbers<\/em>. That is, the argument of the logarithmic function must be greater than zero.<\/p>\r\n<p id=\"fs-id1165137851584\">For example, consider[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x-3\\right).\\,[\/latex]This function is defined for any values of[latex]\\,x\\,[\/latex]such that the argument, in this case[latex]\\,2x-3,[\/latex] is greater than zero. To find the domain, we set up an inequality and solve for[latex]\\,x:[\/latex]<\/p>\r\n\r\n<div id=\"eip-318\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}2x-3&gt;0\\hfill &amp; \\text{Show the argument greater than zero}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x&gt;3\\hfill &amp; \\text{Add 3}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&gt;1.5\\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\text{Divide by 2}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137645047\">In interval notation, the domain of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x-3\\right)\\,[\/latex]is[latex]\\,\\left(1.5,\\infty \\right).[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137423048\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135173951\"><strong>Given a logarithmic function, identify the domain.\r\n<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137823224\" type=\"1\">\r\n \t<li>Set up an inequality showing the argument 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>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\" class=\"solution textbox shaded\">[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 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\">[latex]\\begin{array}{ll}x+3&gt;0\\hfill &amp; \\text{The input must be positive}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&gt;-3\\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\text{Subtract 3}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137638183\">The domain of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)\\,[\/latex]is[latex]\\,\\left(-3,\\infty \\right).[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135692225\" class=\"textbox tryit\">\r\n<h3>Try It<\/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\" class=\"solution textbox shaded\">[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>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\" class=\"solution textbox shaded\">[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 is positive, so this function is defined when[latex]\\,5\u20132x&gt;0.\\,[\/latex]Solving this inequality,<\/p>\r\n\r\n<div id=\"eip-id1165135470032\" class=\"unnumbered\">[latex]\\begin{array}{ll}5-2x&gt;0\\hfill &amp; \\text{The input must be positive}.\\hfill \\\\ \\,\\,\\,-2x&gt;-5\\hfill &amp; \\text{Subtract }5.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&lt;\\frac{5}{2}\\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\text{Divide by }-2\\text{ and switch the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137656879\">The domain of[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(5-2x\\right)\\,[\/latex]is[latex]\\,\\left(\u2013\\infty ,\\frac{5}{2}\\right).[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137874699\" class=\"textbox tryit\">\r\n<h3>Try It<\/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\" class=\"solution textbox shaded\">[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 parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]along with all its transformations: shifts, stretches, compressions, and reflections.<\/p>\r\n<p id=\"fs-id1165137679088\">We begin with the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right).\\,[\/latex]Because every logarithmic function of this form is the inverse of an exponential function with the form[latex]\\,y={b}^{x},[\/latex] their graphs will be reflections of each other across the line[latex]\\,y=x.\\,[\/latex]To illustrate this, we can observe the relationship between the input and output values of[latex]\\,y={2}^{x}\\,[\/latex]and its equivalent[latex]\\,x={\\mathrm{log}}_{2}\\left(y\\right)\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Table_04_04_01\">(Figure)<\/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>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\r\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135509175\">Using the inputs and outputs from <a class=\"autogenerated-content\" href=\"#Table_04_04_01\">(Figure)<\/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\">(Figure)<\/a><strong>.<\/strong><\/p>\r\n\r\n<table id=\"Table_04_04_02\" 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><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(0,1\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(1,2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(3,8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td>[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(1,0\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(2,1\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(4,2\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(8,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137761335\">As we\u2019d expect, the <em>x<\/em>- and <em>y<\/em>-coordinates are reversed for the inverse functions. <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_002\">(Figure)<\/a> shows 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 aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140543\/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. <\/strong>Notice that the graphs of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]and[latex]\\,g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)\\,[\/latex]are reflections about the line[latex]\\,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\n<div id=\"fs-id1165137780760\" class=\"textbox key-takeaways\">\r\n<h3>Characteristics of the Graph of the Parent Function, <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\r\n<p id=\"fs-id1165135520250\">For any real number[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><em>x-<\/em>intercept:[latex]\\,\\left(1,0\\right)\\,[\/latex]and key point [latex]\\left(b,1\\right)[\/latex]<\/li>\r\n \t<li><em>y<\/em>-intercept: none<\/li>\r\n \t<li>increasing if[latex]\\,b&gt;1[\/latex]<\/li>\r\n \t<li>decreasing if[latex]\\,0&lt;b&lt;1[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137464857\">See <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_003\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_003\" class=\"wp-caption aligncenter\">\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\/3252\/2018\/07\/19140545\/CNX_Precalc_Figure_04_04_003G.jpg\" alt=\"\" width=\"824\" height=\"367\" \/> <strong>Figure 3.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137838178\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_004\">(Figure)<\/a> shows how changing the base[latex]\\,b\\,[\/latex]in[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (<em>Note:<\/em> recall that the function[latex]\\,\\mathrm{ln}\\left(x\\right)\\,[\/latex]has base[latex]\\,e\\approx \\text{2}.\\text{718.)}[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_004\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140548\/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 4. <\/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 id=\"fs-id1165137430980\" class=\"bc-section section\">\r\n<div id=\"fs-id1165137871937\" class=\"precalculus howto textbox tryit\">\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 <em>x-<\/em>intercept,[latex]\\,\\left(1,0\\right).[\/latex]<\/li>\r\n \t<li>Plot the key point[latex]\\,\\left(b,1\\right).[\/latex]<\/li>\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n \t<li>State the domain,[latex]\\,\\left(0,\\infty \\right),[\/latex]the range,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]and the vertical asymptote,[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>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\" class=\"solution textbox shaded\">[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 tail of the graph will approach the vertical asymptote[latex]\\,x=0,[\/latex] and the right tail will increase slowly without bound.<\/li>\r\n \t<li>The <em>x<\/em>-intercept is[latex]\\,\\left(1,0\\right).[\/latex]<\/li>\r\n \t<li>The key point[latex]\\,\\left(5,1\\right)\\,[\/latex]is on the graph.<\/li>\r\n \t<li>We 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)<\/a>).<\/li>\r\n<\/ul>\r\n<div id=\"CNX_Precalc_Figure_04_04_005\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140555\/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=\"487\" height=\"367\" \/> <strong>Figure 5.<\/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 the vertical asymptote is[latex]\\,x=0.[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135161300\" class=\"textbox tryit\">\r\n<h3>Try It<\/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 asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137407451\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137407451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137407451\"]<span id=\"fs-id1165134377926\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140603\/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.\" \/><\/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 the vertical asymptote is[latex]\\,x=0.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3><\/h3>\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 other parent functions. We can shift, stretch, compress, and reflect the <span class=\"no-emphasis\">parent function<\/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 <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\r\n<p id=\"fs-id1165135530294\">When a constant[latex]\\,c\\,[\/latex]is added to the input of the parent function[latex]\\,f\\left(x\\right)=lo{g}_{b}\\left(x\\right),[\/latex] the result is a <span class=\"no-emphasis\">horizontal shift<\/span>[latex]\\,c\\,[\/latex]units in the <em>opposite<\/em> direction of the sign on[latex]\\,c.\\,[\/latex]To visualize horizontal shifts, we can observe the general graph of the parent function[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]and for[latex]\\,c&gt;0\\,[\/latex]alongside the shift left,[latex]\\,g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right),[\/latex] and the shift right,[latex]\\,h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right).[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_04_04_007\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_04_04_007\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140605\/CNX_Precalc_Figure_04_04_007n.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.\" width=\"975\" height=\"570\" \/> <strong>Figure 6.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135296307\" class=\"textbox key-takeaways\">\r\n<h3>Horizontal Shifts of the Parent 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 parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]left[latex]\\,c\\,[\/latex]units if[latex]\\,c&gt;0.[\/latex]<\/li>\r\n \t<li>shifts the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]right[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 id=\"fs-id1165137641710\" class=\"precalculus howto textbox tryit\">\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>Identify the horizontal shift:\r\n<ol id=\"fs-id1165137454288\" type=\"a\">\r\n \t<li>If[latex]\\,c&gt;0,[\/latex]shift the graph of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]left[latex]\\,c\\,[\/latex]units.<\/li>\r\n \t<li>If[latex]\\,c&lt;0,[\/latex]shift the graph of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]right[latex]\\,c\\,[\/latex]units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote[latex]\\,x=-c.[\/latex]<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting[latex]\\,c\\,[\/latex]from the[latex]\\,x\\,[\/latex]coordinate.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The Domain is[latex]\\,\\left(-c,\\infty \\right),[\/latex]the range is[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is[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>Graphing a Horizontal Shift of the Parent 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 its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137759883\" class=\"solution textbox shaded\">[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] we notice[latex]\\,x+\\left(-2\\right)=x\u20132.[\/latex]<\/p>\r\n<p id=\"fs-id1165137784630\">Thus[latex]\\,c=-2,[\/latex]so[latex]\\,c&lt;0.\\,[\/latex]This means we will shift the function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\,[\/latex]right 2 units.<\/p>\r\n<p id=\"fs-id1165137836995\">The vertical asymptote is[latex]\\,x=-\\left(-2\\right)\\,[\/latex]or[latex]\\,x=2.[\/latex]<\/p>\r\n<p id=\"fs-id1165134042608\">Consider the three key points from the parent function,[latex]\\,\\left(\\frac{1}{3},-1\\right),[\/latex][latex]\\left(1,0\\right),[\/latex]and[latex]\\,\\left(3,1\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165137475806\">The new coordinates are found by adding 2 to the[latex]\\,x\\,[\/latex]coordinates.<\/p>\r\n<p id=\"fs-id1165137748449\">Label the points[latex]\\,\\left(\\frac{7}{3},-1\\right),[\/latex][latex]\\left(3,0\\right),[\/latex]and[latex]\\,\\left(5,1\\right).[\/latex]<\/p>\r\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 aligncenter\">\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\/3252\/2018\/07\/19140608\/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=\"487\" height=\"363\" \/> <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=\"textbox tryit\">\r\n<h3>Try It<\/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 its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134373485\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134373485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134373485\"]<span id=\"fs-id1165135209395\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140610\/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).\" \/><\/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 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 parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right),[\/latex]the result is a <span class=\"no-emphasis\">vertical shift<\/span>[latex]\\,d\\,[\/latex]units in the direction of the sign on[latex]\\,d.\\,[\/latex]To visualize vertical shifts, we can observe the general graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]alongside the shift up,[latex]\\,g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\,[\/latex]and the shift down,[latex]\\,h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d.[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_04_04_010\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_04_04_010\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140612\/CNX_Precalc_Figure_04_04_010F.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.\" width=\"975\" height=\"741\" \/> <strong>Figure 8.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137767601\" class=\"textbox key-takeaways\">\r\n<h3>Vertical Shifts of the Parent 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 parent 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 parent 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 id=\"fs-id1165137706002\" class=\"precalculus howto textbox tryit\">\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]\\,f\\left(x\\right)={\\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]\\,f\\left(x\\right)={\\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 three key points from the parent function. 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 three points.<\/li>\r\n \t<li>The domain is[latex]\\,\\left(0,\\infty \\right),[\/latex]the range is[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]and the vertical asymptote is[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>\r\n<div id=\"fs-id1165137470059\">\r\n<h3>Graphing a Vertical Shift of the Parent 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 its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137925369\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137925369\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137925369\"]\r\n<p id=\"fs-id1165137465913\">Since the function is[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2,[\/latex]we will notice[latex]\\,d=\u20132.\\,[\/latex]Thus[latex]\\,d&lt;0.[\/latex]<\/p>\r\n<p id=\"fs-id1165135175015\">This means we will shift the function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\,[\/latex]down 2 units.<\/p>\r\n<p id=\"fs-id1165137644429\">The vertical asymptote is[latex]\\,x=0.[\/latex]<\/p>\r\n<p id=\"fs-id1165137408419\">Consider the three key points from the parent function,[latex]\\,\\left(\\frac{1}{3},-1\\right),[\/latex][latex]\\left(1,0\\right),[\/latex]and[latex]\\,\\left(3,1\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165135503945\">The new coordinates are found by subtracting 2 from the <em>y <\/em>coordinates.<\/p>\r\n<p id=\"fs-id1165135421660\">Label the points[latex]\\,\\left(\\frac{1}{3},-3\\right),[\/latex][latex]\\left(1,-2\\right),[\/latex] and[latex]\\,\\left(3,-1\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165135195524\">The domain is[latex]\\,\\left(0,\\infty \\right),[\/latex]the range is[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is[latex]\\,x=0.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_011\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140627\/CNX_Precalc_Figure_04_04_011.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=\"487\" height=\"516\" \/> <strong>Figure 9.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137698285\">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][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135176716\" class=\"textbox tryit\">\r\n<h3>Try It<\/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 its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137874464\" class=\"small\">\r\n<div class=\"textbox shaded\">[reveal-answer q=\"676362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"676362\"]<span id=\"fs-id1165137874471\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140633\/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).\" \/><\/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[\/hidden-answer]\r\n\r\n<\/div>\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 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 parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is multiplied by a constant[latex]\\,a&gt;0,[\/latex] the result is a <span class=\"no-emphasis\">vertical stretch<\/span> or <span class=\"no-emphasis\">compression<\/span> of the original graph. To visualize stretches and compressions, we set[latex]\\,a&gt;1\\,[\/latex]and observe the general graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]alongside the vertical stretch,[latex]\\,g\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]and the vertical compression,[latex]\\,h\\left(x\\right)=\\frac{1}{a}{\\mathrm{log}}_{b}\\left(x\\right).[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_04_04_013\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_04_04_013\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140638\/CNX_Precalc_Figure_04_04_013n.jpg\" alt=\"\" width=\"975\" height=\"758\" \/> <strong>Figure 10.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137433996\" class=\"textbox key-takeaways\">\r\n<h3>Vertical Stretches and Compressions of the Parent 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 parent 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 parent 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 <em>x<\/em>-intercept[latex]\\,\\left(1,0\\right).[\/latex]<\/li>\r\n \t<li>has domain[latex]\\,\\left(0,\\infty \\right).[\/latex]<\/li>\r\n \t<li>has range[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"precalculus howto\">\r\n<p id=\"fs-id1165135169307\"><strong>Given a logarithmic function with the form[latex]\\,f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right),[\/latex][latex]a&gt;0,[\/latex]graph the translation.<\/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]\\,f\\left(x\\right)={\\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]\\,f\\left(x\\right)={\\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 three key points from the parent function. Find new coordinates for the shifted functions by multiplying the[latex]\\,y\\,[\/latex]coordinates by[latex]\\,a.[\/latex]<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The domain is[latex]\\,\\left(0,\\infty \\right),[\/latex]the range is[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]and the vertical asymptote is[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>Graphing a Stretch or Compression of the Parent 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 its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"420184\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"420184\"]Since the function is[latex]\\,f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right),[\/latex]we will notice[latex]\\,a=2.[\/latex]\r\n<p id=\"fs-id1165135384321\">This means we will stretch the function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\,[\/latex]by a factor of 2.<\/p>\r\n<p id=\"fs-id1165135481989\">The vertical asymptote is[latex]\\,x=0.[\/latex]<\/p>\r\n<p id=\"fs-id1165137757801\">Consider the three key points from the parent function,[latex]\\,\\left(\\frac{1}{4},-1\\right),[\/latex][latex]\\left(1,0\\right),\\,[\/latex]and[latex]\\,\\left(4,1\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165135570058\">The new coordinates are found by multiplying the[latex]\\,y\\,[\/latex]coordinates by 2.<\/p>\r\n<p id=\"fs-id1165137837989\">Label the points[latex]\\,\\left(\\frac{1}{4},-2\\right),[\/latex][latex]\\left(1,0\\right)\\,,[\/latex] and[latex]\\,\\left(4,\\text{2}\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165135543469\">The domain is[latex]\\,\\left(0,\\,\\infty \\right),[\/latex] the range is[latex]\\,\\left(-\\infty ,\\infty \\right),\\,[\/latex]and the vertical asymptote is[latex]\\,x=0.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2455&amp;action=edit#CNX_Precalc_Figure_04_04_014\">(Figure)<\/a><strong>.<\/strong><\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_014\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140641\/CNX_Precalc_Figure_04_04_014.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).\" width=\"487\" height=\"366\" \/> <strong>Figure 11.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135566827\">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[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134079655\" class=\"textbox tryit\">\r\n<h3>Try It<\/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}\\,{\\mathrm{log}}_{4}\\left(x\\right)\\,[\/latex]alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137778952\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137778952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137778952\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140643\/CNX_Precalc_Figure_04_04_015.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1\/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).\" \/>\r\n<p id=\"fs-id1165137768488\">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[\/hidden-answer]\r\n\r\n<\/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>Combining a Shift and a Stretch<\/h3>\r\n<p id=\"fs-id1165137863045\">Sketch a graph of[latex]\\,f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\right).\\,[\/latex]State the domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137935559\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137935559\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137935559\"]\r\n<p id=\"fs-id1165137935561\">Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_016\">(Figure)<\/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 aligncenter\">\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\/3252\/2018\/07\/19140646\/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=\"487\" height=\"441\" \/> <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][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135560730\" class=\"textbox tryit\">\r\n<h3>Try It<\/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 asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137437228\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137437228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137437228\"]<span id=\"fs-id1165135177663\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140649\/CNX_Precalc_Figure_04_04_017.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" \/><\/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]<\/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 <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\r\n<p id=\"fs-id1165135169315\">When the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is multiplied by[latex]\\,-1,[\/latex]the result is a <span class=\"no-emphasis\">reflection<\/span> 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 parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]alongside the reflection about the <em>x<\/em>-axis,[latex]\\,g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)\\,[\/latex]and the reflection about the <em>y<\/em>-axis,[latex]\\,h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_04_04_018\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140655\/CNX_Precalc_Figure_04_04_018n.jpg\" alt=\"\" width=\"975\" height=\"786\" \/> <strong>Figure 13.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135190744\" class=\"textbox key-takeaways\">\r\n<h3>Reflections of the Parent 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 id=\"fs-id1165137832285\">\r\n \t<li>reflects the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]about the <em>x<\/em>-axis.<\/li>\r\n \t<li>has domain,[latex]\\,\\left(0,\\infty \\right),[\/latex] range,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] and vertical asymptote,[latex]\\,x=0,[\/latex] which are unchanged from the parent 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 parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]about the <em>y<\/em>-axis.<\/li>\r\n \t<li>has domain[latex]\\,\\left(-\\infty ,0\\right).[\/latex]<\/li>\r\n \t<li>has range,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] and vertical asymptote,[latex]\\,x=0,[\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"precalculus howto\">\r\n<p id=\"fs-id1165137638837\"><strong>Given a logarithmic function with the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] graph a translation.<\/strong><\/p>\r\n\r\n<table id=\"Table_04_04_08\" class=\"unnumbered\" summary=\"The first column gives the following instructions of graphing a translation of f(x)=-log_b(x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the x-axis; 4. Draw a smooth curve through the points; 5. State the domain, (0, infinity), the range, (-infinity, infinity), and the vertical asymptote x=0. The second column gives the following instructions of graphing a translation of f(x)=log_b(-x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (-1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the y-axis; 4. Draw a smooth curve through the points; 5. State the domain, (-infinity, 0), the range, (-infinity, infinity), and the vertical asymptote x=0.\">\r\n<thead>\r\n<tr>\r\n<th>[latex]\\text{If }f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n<th>[latex]\\text{If }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>\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>\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>\r\n<td>\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>\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>\r\n<td>\r\n<ol id=\"fs-id1165135195408\" start=\"3\" type=\"1\">\r\n \t<li>Reflect the graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]about the <em>x<\/em>-axis.<\/li>\r\n<\/ol>\r\n<\/td>\r\n<td>\r\n<ol start=\"3\" type=\"1\">\r\n \t<li>Reflect the graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]about the <em>y<\/em>-axis.<\/li>\r\n<\/ol>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\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>\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>\r\n<td>\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>\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>Graphing a Reflection of a Logarithmic Function<\/h3>\r\n<p id=\"fs-id1165137849038\">Sketch a graph of[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)\\,[\/latex]alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137836523\" class=\"solution textbox shaded\">[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 parent function is increasing. Since the <em>input<\/em> value is multiplied by[latex]\\,-1,[\/latex][latex]f\\,[\/latex]is a reflection of the parent graph about the <em>y-<\/em>axis. Thus,[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)\\,[\/latex]will be decreasing as[latex]\\,x\\,[\/latex]moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote[latex]\\,x=0.\\,[\/latex]<\/li>\r\n \t<li>The <em>x<\/em>-intercept is[latex]\\,\\left(-1,0\\right).[\/latex]<\/li>\r\n \t<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\r\n<\/ul>\r\n<div id=\"CNX_Precalc_Figure_04_04_019\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140657\/CNX_Precalc_Figure_04_04_019.jpg\" alt=\"Graph of two functions. The parent function is y=log(x), with an asymptote at x=0 and labeled points at (1, 0), and (10, 0).The translation function f(x)=log(-x) has an asymptote at x=0 and labeled points at (-1, 0) and (-10, 1).\" width=\"487\" height=\"363\" \/> <strong>Figure 14.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165134042202\">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][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135650792\" class=\"textbox tryit\">\r\n<h3>Try It<\/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 asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137836698\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137836698\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137836698\"]<span id=\"fs-id1165137855148\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140706\/CNX_Precalc_Figure_04_04_020.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" \/><\/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]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134579621\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165134579627\"><strong>Given a logarithmic equation, use a graphing calculator to approximate solutions.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137431118\" type=\"1\">\r\n \t<li>Press <strong>[Y=]<\/strong>. Enter the given logarithm equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\r\n \t<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\r\n \t<li>To find the value of[latex]\\,x,[\/latex] we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \u201cintersect\u201d and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of[latex]\\,x,[\/latex]for the point(s) of intersection.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_04_09\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135414229\">\r\n<div id=\"fs-id1165135414231\">\r\n<h3>Approximating the Solution of a Logarithmic Equation<\/h3>\r\n<p id=\"fs-id1165135414236\">Solve[latex]\\,4\\mathrm{ln}\\left(x\\right)+1=-2\\mathrm{ln}\\left(x-1\\right)\\,[\/latex]graphically. Round to the nearest thousandth.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135193432\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135193432\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135193432\"]Press <strong>[Y=]<\/strong> and enter[latex]\\,4\\mathrm{ln}\\left(x\\right)+1\\,[\/latex]next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter[latex]\\,-2\\mathrm{ln}\\left(x-1\\right)\\,[\/latex]next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for[latex]\\,x\\,[\/latex]and \u201310 to 10 for[latex]\\,y.\\,[\/latex]Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to right of[latex]\\,x=1.[\/latex]\r\n<p id=\"fs-id1165135245763\">For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em>x<\/em>-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) So, to the nearest thousandth,[latex]\\,x\\approx 1.339.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137891344\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_04_04_09\">\r\n<div id=\"fs-id1165137639529\">\r\n<p id=\"fs-id1165137639531\">Solve[latex]\\,5\\mathrm{log}\\left(x+2\\right)=4-\\mathrm{log}\\left(x\\right)\\,[\/latex]graphically. Round to the nearest thousandth.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137817452\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137817452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137817452\"]\r\n<p id=\"fs-id1165137817454\">[latex]x\\approx 3.049[\/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-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\">(Figure)<\/a> to arrive at the general equation for translating exponential functions.\r\n<table id=\"Table_04_04_009\"><caption>\u00a0<\/caption>\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\">Translations of the Parent Function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Translation<\/th>\r\n<th>Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Shift\r\n<ul id=\"fs-id1165137416971\">\r\n \t<li>Horizontally[latex]\\,c\\,[\/latex]units to the left<\/li>\r\n \t<li>Vertically[latex]\\,d\\,[\/latex]units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Stretch and Compress\r\n<ul id=\"fs-id1165137427553\">\r\n \t<li>Stretch if[latex]\\,|a|&gt;1[\/latex]<\/li>\r\n \t<li>Compression if[latex]\\,|a|&lt;1[\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em>x<\/em>-axis<\/td>\r\n<td>[latex]y=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em>y<\/em>-axis<\/td>\r\n<td>[latex]y={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>General equation for all translations<\/td>\r\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1165137414493\" class=\"textbox key-takeaways\">\r\n<h3>Translations of Logarithmic Functions<\/h3>\r\n<p id=\"fs-id1165137414501\">All translations of the parent logarithmic function,[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] have the form<\/p>\r\n\r\n<div id=\"fs-id1165135408512\">[latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/div>\r\n<p id=\"fs-id1165137734655\">where the parent function,[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right),b&gt;1,[\/latex]is<\/p>\r\n\r\n<ul>\r\n \t<li>shifted vertically up[latex]\\,d\\,[\/latex]units.<\/li>\r\n \t<li>shifted horizontally to the left[latex]\\,c\\,[\/latex]units.<\/li>\r\n \t<li>stretched vertically by a factor of[latex]\\,|a|\\,[\/latex]if[latex]\\,|a|&gt;0.[\/latex]<\/li>\r\n \t<li>compressed vertically by a factor of[latex]\\,|a|\\,[\/latex]if[latex]\\,0&lt;|a|&lt;1.[\/latex]<\/li>\r\n \t<li>reflected about the <em>x-<\/em>axis when[latex]\\,a&lt;0.[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137725084\">For[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(-x\\right),[\/latex] the graph of the parent function is reflected about the <em>y<\/em>-axis.<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_04_04_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135296269\">\r\n<div id=\"fs-id1165135296271\">\r\n<h3>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\" class=\"solution textbox shaded\">[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 at[latex]\\,x=-4.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137871955\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165137871960\">The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to[latex]\\,x=-4.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137748682\" class=\"textbox tryit\">\r\n<h3>Try It<\/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\" class=\"solution textbox shaded\">[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>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)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_021\" class=\"small aligncenter\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140708\/CNX_Precalc_Figure_04_04_021.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).\" width=\"487\" height=\"367\" \/> <strong>Figure 15.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135342977\" class=\"solution textbox shaded\">[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=\u20132\\,[\/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\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+k[\/latex]<\/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,\u20131\\right).\\,[\/latex]Substituting[latex]\\,\\left(\u20131,1\\right),[\/latex]<\/p>\r\n\r\n<div id=\"eip-id1165134101923\" class=\"unnumbered\">[latex]\\begin{array}{ll}1=-a\\mathrm{log}\\left(-1+2\\right)+k\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\hfill &amp; \\text{Substitute }\\left(-1,1\\right).\\hfill \\\\ 1=-a\\mathrm{log}\\left(1\\right)+k\\hfill &amp; \\text{Arithmetic}.\\hfill \\\\ 1=k\\hfill &amp; \\text{log(1)}=0.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137628655\">Next, substituting in[latex]\\,\\left(2,\u20131\\right)[\/latex],<\/p>\r\n\r\n<div id=\"eip-id1165135431720\" class=\"unnumbered\">[latex]\\begin{array}{lll}-1=-a\\mathrm{log}\\left(2+2\\right)+1\\hfill &amp; \\hfill &amp; \\text{Plug in }\\left(2,-1\\right).\\hfill \\\\ -2=-a\\mathrm{log}\\left(4\\right)\\hfill &amp; \\hfill &amp; \\text{Arithmetic}.\\hfill \\\\ \\text{ }a=\\frac{2}{\\mathrm{log}\\left(4\\right)}\\hfill &amp; \\hfill &amp; \\text{Solve for }a.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165135192211\">This gives us the equation[latex]\\,f\\left(x\\right)=\u2013\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137735581\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165137735586\">We can verify this answer by comparing the function values in <a class=\"autogenerated-content\" href=\"#Table_04_04_010\">(Figure)<\/a> with the points on the graph in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_021\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"Table_04_04_010\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<td>\u22120.58496<\/td>\r\n<td>\u22121<\/td>\r\n<td>\u22121.3219<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td>\u22121.5850<\/td>\r\n<td>\u22121.8074<\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22122.1699<\/td>\r\n<td>\u22122.3219<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"eip-id1525883\" class=\"textbox tryit\">\r\n<h3>Try It<\/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)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_022\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140711\/CNX_Precalc_Figure_04_04_022.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).\" width=\"487\" height=\"442\" \/> <strong>Figure 16.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137436435\" class=\"solution textbox shaded\">[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 textbox shaded\">\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)<\/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\\,|\\,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:\/\/openstaxcollege.org\/l\/graphexplog\">Graph an Exponential Function and Logarithmic Function<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/matchexplog\">Match Graphs with Exponential and Logarithmic Functions<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.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>General Form for the Translation of the Parent Logarithmic Function[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<td>[latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/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 argument greater than zero, and solve for[latex]\\,x.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_04_01\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_04_04_02\">(Figure)<\/a><\/li>\r\n \t<li>The graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]has an <em>x-<\/em>intercept at[latex]\\,\\left(1,0\\right),[\/latex]domain[latex]\\,\\left(0,\\infty \\right),[\/latex]range[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]vertical asymptote[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&lt;1,[\/latex] the function is decreasing.<\/li>\r\n<\/ul>\r\nSee <a class=\"autogenerated-content\" href=\"#Example_04_04_03\">(Figure)<\/a>.<\/li>\r\n \t<li>The equation[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)\\,[\/latex]shifts the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]horizontally\r\n<ul id=\"fs-id1165135512562\">\r\n \t<li>left[latex]\\,c\\,[\/latex]units if[latex]\\,c&gt;0.[\/latex]<\/li>\r\n \t<li>right[latex]\\,c\\,[\/latex]units if[latex]\\,c&lt;0.[\/latex]<\/li>\r\n<\/ul>\r\nSee <a class=\"autogenerated-content\" href=\"#Example_04_04_04\">(Figure)<\/a>.<\/li>\r\n \t<li>The equation[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\,[\/latex]shifts the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]vertically\r\n<ul id=\"fs-id1165137761068\">\r\n \t<li>up[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\nSee <a class=\"autogenerated-content\" href=\"#Example_04_04_05\">(Figure)<\/a>.<\/li>\r\n \t<li>For any constant[latex]\\,a&gt;0,[\/latex] the equation[latex]\\,f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\r\n<ul id=\"fs-id1165134040579\">\r\n \t<li>stretches the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]vertically by a factor of[latex]\\,a\\,[\/latex]if[latex]\\,|a|&gt;1.[\/latex]<\/li>\r\n \t<li>compresses the parent 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\nSee <a class=\"autogenerated-content\" href=\"#Example_04_04_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_04_04_07\">(Figure)<\/a>.<\/li>\r\n \t<li>When the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is multiplied by[latex]\\,-1,[\/latex] the result is a reflection about the <em>x<\/em>-axis. When the input is multiplied by[latex]\\,-1,[\/latex] the result is a reflection about the <em>y<\/em>-axis.\r\n<ul id=\"fs-id1165135186594\">\r\n \t<li>The equation[latex]\\,f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]represents a reflection of the parent function about the <em>x-<\/em>axis.<\/li>\r\n \t<li>The equation[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)\\,[\/latex]represents a reflection of the parent function about the <em>y-<\/em>axis.<\/li>\r\n<\/ul>\r\nSee <a class=\"autogenerated-content\" href=\"#Example_04_04_08\">(Figure)<\/a>.\r\n<ul id=\"fs-id1165137834414\">\r\n \t<li>A graphing calculator may be used to approximate solutions to some logarithmic equations See <a class=\"autogenerated-content\" href=\"#Example_04_04_09\">(Figure)<\/a>.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>All translations of the logarithmic function can be summarized by the general equation[latex]\\, f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Table_04_04_009\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_04_04_10\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_04_04_11\">(Figure)<\/a>.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135189917\" class=\"textbox exercises\">\r\n<h3>Section Exercises<\/h3>\r\n<div id=\"fs-id1165135189921\" class=\"bc-section section\">\r\n<h4>Verbal<\/h4>\r\n<div id=\"fs-id1165135582182\">\r\n<div id=\"fs-id1165135582184\">\r\n<p id=\"fs-id1165135582187\">The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135582193\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135582193\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135582193\"]\r\n<p id=\"fs-id1165135582195\">Since the functions are inverses, their graphs are mirror images about the line[latex]\\,y=x.\\,[\/latex]So for every point[latex]\\,\\left(a,b\\right)\\,[\/latex]on the graph of a logarithmic function, there is a corresponding point[latex]\\,\\left(b,a\\right)\\,[\/latex]on the graph of its inverse exponential function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137855282\">\r\n<div id=\"fs-id1165137855284\">\r\n<p id=\"fs-id1165137855286\">What type(s) of translation(s), if any, affect the range of a logarithmic function?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137855292\">\r\n<div id=\"fs-id1165137855294\">\r\n<p id=\"fs-id1165137855296\">What type(s) of translation(s), if any, affect the domain of a logarithmic function?<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"917821\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"917821\"]Shifting the function right or left and reflecting the function about the y-axis will affect its domain.[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165137424694\">\r\n<p id=\"fs-id1165137424697\">Consider the general logarithmic function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right).\\,[\/latex]Why can\u2019t[latex]\\,x\\,[\/latex]be zero?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n<p id=\"fs-id1165137697130\">Does the graph of a general logarithmic function have a horizontal asymptote? Explain.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137459912\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137459912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137459912\"]\r\n<p id=\"fs-id1165137459914\">No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137459922\" class=\"bc-section section\">\r\n<h4>Algebraic<\/h4>\r\n<p id=\"fs-id1165137459927\">For the following exercises, state the domain and range of the function.<\/p>\r\n\r\n<div id=\"fs-id1165137459930\">\r\n<div id=\"fs-id1165137737591\">[latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135264771\">\r\n<div id=\"fs-id1165135264773\">\r\n<p id=\"fs-id1165135264775\">[latex]h\\left(x\\right)=\\mathrm{ln}\\left(\\frac{1}{2}-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135257219\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135257219\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135257219\"]\r\n<p id=\"fs-id1165135257221\">Domain:[latex]\\,\\left(-\\infty ,\\frac{1}{2}\\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135253806\">\r\n<div id=\"fs-id1165135253808\">\r\n<p id=\"fs-id1165135253810\">[latex]g\\left(x\\right)={\\mathrm{log}}_{5}\\left(2x+9\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135415813\">\r\n<div id=\"fs-id1165135415815\">\r\n<p id=\"fs-id1165135415817\">[latex]h\\left(x\\right)=\\mathrm{ln}\\left(4x+17\\right)-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137641046\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137641046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137641046\"]\r\n<p id=\"fs-id1165137641048\">Domain:[latex]\\,\\left(-\\frac{17}{4},\\infty \\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137843712\">\r\n<div id=\"fs-id1165137843714\">\r\n<p id=\"fs-id1165137843716\">[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(12-3x\\right)-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135503696\">For the following exercises, state the domain and the vertical asymptote of the function.<\/p>\r\n\r\n<div id=\"fs-id1165134037565\">\r\n<div id=\"fs-id1165134037567\">\r\n<p id=\"fs-id1165134037569\">[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-5\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137771436\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137771436\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137771436\"]\r\n<p id=\"fs-id1165137771438\">Domain:[latex]\\,\\left(5,\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137933908\">\r\n<div id=\"fs-id1165137933910\">\r\n<p id=\"fs-id1165137933912\">[latex]\\,g\\left(x\\right)=\\mathrm{ln}\\left(3-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137737027\">\r\n<div id=\"fs-id1165137737029\">\r\n<p id=\"fs-id1165137737031\">[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(3x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135256140\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135256140\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135256140\"]\r\n<p id=\"fs-id1165135256142\">Domain:[latex]\\,\\left(-\\frac{1}{3},\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=-\\frac{1}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135188081\">\r\n<div id=\"fs-id1165135188083\">\r\n<p id=\"fs-id1165135188086\">[latex]\\,f\\left(x\\right)=3\\mathrm{log}\\left(-x\\right)+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134188310\">\r\n<div id=\"fs-id1165134188313\">\r\n<p id=\"fs-id1165134188315\">[latex]\\,g\\left(x\\right)=-\\mathrm{ln}\\left(3x+9\\right)-7[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135532384\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135532384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135532384\"]\r\n<p id=\"fs-id1165135208715\">Domain:[latex]\\,\\left(-3,\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=-3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135194730\">For the following exercises, state the domain, vertical asymptote, and end behavior of the function.<\/p>\r\n\r\n<div id=\"fs-id1165135194734\">\r\n<div id=\"fs-id1165135194736\">\r\n<p id=\"fs-id1165135194738\">[latex]f\\left(x\\right)=\\mathrm{ln}\\left(2-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137427662\">\r\n<div id=\"fs-id1165137427665\">\r\n<p id=\"fs-id1165137427667\">[latex]f\\left(x\\right)=\\mathrm{log}\\left(x-\\frac{3}{7}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137715430\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137715430\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137715430\"]Domain: [latex]\\left(\\frac{3}{7},\\infty \\right)[\/latex];[\/hidden-answer]<\/div>\r\n<div id=\"fs-id1165137431240\">\r\n<div id=\"fs-id1165137431242\">\r\n<p id=\"fs-id1165137431244\">[latex]h\\left(x\\right)=-\\mathrm{log}\\left(3x-4\\right)+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134194950\">\r\n<div id=\"fs-id1165134194952\">\r\n<p id=\"fs-id1165134194954\">[latex]g\\left(x\\right)=\\mathrm{ln}\\left(2x+6\\right)-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135358044\" class=\"solution textbox shaded\">\r\n\r\n[reveal-answer q=\"fs-id1165135358044\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"fs-id1165135358044\"]\r\n\r\nDomain: [latex]\\left(-3,\\infty \\right)[\/latex]; Vertical asymptote: [latex]x=-3[\/latex];[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135532457\">\r\n<div id=\"fs-id1165135532459\">\r\n<p id=\"fs-id1165135532461\">[latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(15-5x\\right)+6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137426354\">For the following exercises, state the domain, range, and <em>x<\/em>- and <em>y<\/em>-intercepts, if they exist. If they do not exist, write DNE.<\/p>\r\n\r\n<div id=\"fs-id1165137426368\">\r\n<div id=\"fs-id1165137426370\">\r\n<p id=\"fs-id1165137426372\">[latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x-1\\right)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165132983045\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132983045\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132983045\"]\r\n<p id=\"fs-id1165132983048\">Domain:[latex]\\,\\left(1,\\infty \\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=1;\\,[\/latex]<em>x<\/em>-intercept:[latex]\\,\\left(\\frac{5}{4},0\\right);\\,[\/latex]<em>y<\/em>-intercept: DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137755640\">\r\n<div id=\"fs-id1165135543374\">\r\n<p id=\"fs-id1165135543376\">[latex]f\\left(x\\right)=\\mathrm{log}\\left(5x+10\\right)+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135210139\">\r\n<div id=\"fs-id1165135210141\">\r\n<p id=\"fs-id1165135210143\">[latex]g\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137426893\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137426893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137426893\"]\r\n<p id=\"fs-id1165137426895\">Domain:[latex]\\,\\left(-\\infty ,0\\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=0;\\,[\/latex]<em>x<\/em>-intercept:[latex]\\,\\left(-{e}^{2},0\\right);\\,[\/latex]<em>y<\/em>-intercept: DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137810010\">\r\n<div id=\"fs-id1165137810012\">\r\n<p id=\"fs-id1165135401817\">[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137942453\">\r\n<div id=\"fs-id1165137942455\">[latex]h\\left(x\\right)=3\\mathrm{ln}\\left(x\\right)-9[\/latex]<\/div>\r\n<div id=\"fs-id1165135186502\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135186502\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135186502\"]\r\n<p id=\"fs-id1165135186504\">Domain:[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex] Vertical asymptote: [latex]\\,x=0;\\,[\/latex]<em>x<\/em>-intercept:[latex]\\,\\left({e}^{3},0\\right);\\,[\/latex]<em>y<\/em>-intercept: DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137422264\" class=\"bc-section section\">\r\n<h4>Graphical<\/h4>\r\n<p id=\"fs-id1165137422269\">For the following exercises, match each function in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_201\">(Figure)<\/a> with the letter corresponding to its graph.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_201\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140718\/CNX_PreCalc_Figure_04_04_201.jpg\" alt=\"Graph of five logarithmic functions.\" width=\"487\" height=\"440\" \/> <strong>Figure 17.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134032273\">\r\n<div id=\"fs-id1165134032275\">\r\n<p id=\"fs-id1165134032277\">[latex]d\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135194197\">\r\n<div id=\"fs-id1165137911558\">\r\n<p id=\"fs-id1165137911560\">[latex]f\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134255558\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134255558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134255558\"]\r\n<p id=\"fs-id1165134255561\">B<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134255566\">\r\n<div id=\"fs-id1165134255568\">\r\n<p id=\"fs-id1165134255570\">[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137758154\">\r\n<div id=\"fs-id1165137758156\">\r\n<p id=\"fs-id1165137758158\">[latex]h\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165135571660\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135571660\"]\r\n<p id=\"fs-id1165135571660\">C<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135571665\">\r\n<div>\r\n<p id=\"fs-id1165135571669\">[latex]j\\left(x\\right)={\\mathrm{log}}_{25}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, match each function in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_202\">(Figure)<\/a> with the letter corresponding to its graph.\r\n<div id=\"CNX_Precalc_Figure_04_04_202\" class=\"small aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"342\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140728\/CNX_PreCalc_Figure_04_04_202.jpg\" alt=\"Graph of three logarithmic functions.\" width=\"342\" height=\"440\" \/> <strong>Figure 18.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134380389\">\r\n<div id=\"fs-id1165135400154\">\r\n<p id=\"fs-id1165135400156\">[latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{3}}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135206082\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135206082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135206082\"]B[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135206089\">\r\n<div id=\"fs-id1165135206092\">\r\n<p id=\"fs-id1165135206094\">[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137855381\">\r\n<div id=\"fs-id1165137855383\">\r\n<p id=\"fs-id1165137855385\">[latex]h\\left(x\\right)={\\mathrm{log}}_{\\frac{3}{4}}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137637948\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137637948\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137637948\"]\r\n<p id=\"fs-id1165135394336\">C<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"eip-967\">For the following exercises, sketch the graphs of each pair of functions on the same axis.<\/p>\r\n\r\n<div id=\"fs-id1165135394346\">\r\n<div id=\"fs-id1165135394348\">\r\n<p id=\"fs-id1165135394350\">[latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)={10}^{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137862457\">\r\n<div id=\"fs-id1165137862459\">\r\n<p id=\"fs-id1165137862461\">[latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"809684\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"809684\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140735\/CNX_PreCalc_Figure_04_04_204.jpg\" alt=\"Graph of two functions, g(x) = log_(1\/2)(x) in orange and f(x)=log(x) in blue.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135180056\">\r\n<div id=\"fs-id1165135180058\">\r\n<p id=\"fs-id1165135180060\">[latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135536589\">\r\n<div id=\"fs-id1165134325846\">\r\n<p id=\"fs-id1165134325848\">[latex]f\\left(x\\right)={e}^{x}\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135496605\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135496605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135496605\"]<span id=\"fs-id1165135496611\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140747\/CNX_PreCalc_Figure_04_04_206.jpg\" alt=\"Graph of two functions, g(x) = ln(1\/2)(x) in orange and f(x)=e^(x) in blue.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135496626\">For the following exercises, match each function in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_207\">(Figure)<\/a> with the letter corresponding to its graph.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_04_207\" class=\"small aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"374\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140759\/CNX_Precalc_Figure_04_04_207.jpg\" alt=\"Graph of three logarithmic functions.\" width=\"374\" height=\"377\" \/> <strong>Figure 19.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135189876\">\r\n<div id=\"fs-id1165135189878\">\r\n<p id=\"fs-id1165135189880\">[latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(-x+2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137656835\">\r\n<div id=\"fs-id1165137656837\">\r\n<p id=\"fs-id1165137656839\">[latex]g\\left(x\\right)=-{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<div id=\"fs-id1165137656835\">\r\n<div>\r\n\r\n[reveal-answer q=\"478955\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"478955\"]C[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135632110\">\r\n<div id=\"fs-id1165135632112\">\r\n<p id=\"fs-id1165135408522\">[latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, sketch the graph of the indicated function.\r\n<div id=\"fs-id1165135571798\">\r\n<div id=\"fs-id1165135571800\">\r\n<p id=\"fs-id1165135571802\">[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"eip-id2073097\">\r\n<div class=\"textbox shaded\">[reveal-answer q=\"916976\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"916976\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140801\/CNX_PreCalc_Figure_04_04_208.jpg\" alt=\"Graph of f(x)=log_2(x+2).\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137849072\">\r\n<div id=\"fs-id1165137849074\">\r\n<p id=\"fs-id1165137849076\">[latex]\\,f\\left(x\\right)=2\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135251375\">\r\n<div id=\"fs-id1165135251377\">\r\n<p id=\"fs-id1165135251380\">[latex]\\,f\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134032402\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134032402\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134032402\"]<span id=\"fs-id1165134032408\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140809\/CNX_PreCalc_Figure_04_04_210.jpg\" alt=\"Graph of f(x)=ln(-x).\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134032422\">\r\n<div id=\"fs-id1165134032425\">\r\n<p id=\"fs-id1165134032427\">[latex]g\\left(x\\right)=\\mathrm{log}\\left(4x+16\\right)+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137932668\">\r\n<div id=\"fs-id1165137932670\">\r\n<p id=\"fs-id1165137932672\">[latex]g\\left(x\\right)=\\mathrm{log}\\left(6-3x\\right)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137705397\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137705397\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137705397\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140814\/CNX_PreCalc_Figure_04_04_212.jpg\" alt=\"Graph of g(x)=log(6-3x)+1.\" \/>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137705417\">\r\n<div id=\"fs-id1165137705419\">\r\n<p id=\"fs-id1165137563331\">[latex]h\\left(x\\right)=-\\frac{1}{2}\\mathrm{ln}\\left(x+1\\right)-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135439867\">For the following exercises, write a logarithmic equation corresponding to the graph shown.<\/p>\r\n\r\n<div id=\"fs-id1165135439871\">\r\n<div id=\"fs-id1165135439873\">\r\n<p id=\"fs-id1165135439875\">Use[latex]\\,y={\\mathrm{log}}_{2}\\left(x\\right)\\,[\/latex]as the parent function.<\/p>\r\n<span id=\"fs-id1165135443953\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140818\/CNX_PreCalc_Figure_04_04_214.jpg\" alt=\"The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134360851\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134360851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134360851\"]\r\n<p id=\"fs-id1165134360853\">[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{2}\\left(-\\left(x-1\\right)\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135191066\">\r\n<div id=\"fs-id1165135191068\">\r\n<p id=\"fs-id1165135191070\">Use[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\,[\/latex]as the parent function.<\/p>\r\n<span id=\"fs-id1165137674237\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140830\/CNX_PreCalc_Figure_04_04_215.jpg\" alt=\"The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137674252\">\r\n<div id=\"fs-id1165137674255\">\r\n<p id=\"fs-id1165137674257\">Use[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\,[\/latex]as the parent function.<\/p>\r\n<span id=\"fs-id1165134086070\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140833\/CNX_PreCalc_Figure_04_04_216.jpg\" alt=\"The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.\" \/><\/span>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134086084\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134086084\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134086084\"]\r\n<p id=\"fs-id1165134086086\">[latex]f\\left(x\\right)=3{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135203463\">\r\n<div id=\"fs-id1165135203465\">\r\n<p id=\"fs-id1165135203467\">Use[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)\\,[\/latex]as the parent function.<\/p>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140840\/CNX_PreCalc_Figure_04_04_217.jpg\" alt=\"The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137705205\" class=\"bc-section section\">\r\n<h4>Technology<\/h4>\r\n<p id=\"fs-id1165137705211\">For the following exercises, use a graphing calculator to find approximate solutions to each equation.<\/p>\r\n\r\n<div id=\"fs-id1165137705215\">\r\n<div>\r\n<p id=\"fs-id1165135547452\">[latex]\\mathrm{log}\\left(x-1\\right)+2=\\mathrm{ln}\\left(x-1\\right)+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137674291\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137674291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137674291\"]\r\n<p id=\"fs-id1165137674294\">[latex]x=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137674311\">\r\n<div id=\"fs-id1165137674313\">\r\n<p id=\"fs-id1165135438432\">[latex]\\mathrm{log}\\left(2x-3\\right)+2=-\\mathrm{log}\\left(2x-3\\right)+5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1165135176229\">\r\n<p id=\"fs-id1165135176231\">[latex]\\mathrm{ln}\\left(x-2\\right)=-\\mathrm{ln}\\left(x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135337765\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135337765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135337765\"]\r\n<p id=\"fs-id1165135181772\">[latex]x\\approx \\text{2}\\text{.303}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135181791\">\r\n<div id=\"fs-id1165135181793\">\r\n<p id=\"fs-id1165135181795\">[latex]2\\mathrm{ln}\\left(5x+1\\right)=\\frac{1}{2}\\mathrm{ln}\\left(-5x\\right)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135367864\">\r\n<div id=\"fs-id1165135367866\">\r\n<p id=\"fs-id1165135367869\">[latex]\\frac{1}{3}\\mathrm{log}\\left(1-x\\right)=\\mathrm{log}\\left(x+1\\right)+\\frac{1}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135174505\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135174505\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135174505\"]\r\n<p id=\"fs-id1165135485716\">[latex]x\\approx -0.472[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135485736\" class=\"bc-section section\">\r\n<h4>Extensions<\/h4>\r\n<div>\r\n<div id=\"fs-id1165135485744\">\r\n<p id=\"fs-id1165135485746\">Let[latex]\\,b\\,[\/latex]be any positive real number such that[latex]\\,b\\ne 1.\\,[\/latex]What must[latex]\\,{\\mathrm{log}}_{b}1\\,[\/latex]be equal to? Verify the result.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135193274\">\r\n<div id=\"fs-id1165135193276\">\r\n<p id=\"fs-id1165135193279\">Explore and discuss the graphs of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right).\\,[\/latex]Make a conjecture based on the result.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134151930\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134151930\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134151930\"]\r\n<p id=\"fs-id1165134151932\">The graphs of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right)\\,[\/latex]appear to be the same; Conjecture: for any positive base[latex]\\,b\\ne 1,[\/latex][latex]\\,{\\mathrm{log}}_{b}\\left(x\\right)=-{\\mathrm{log}}_{\\frac{1}{b}}\\left(x\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135369592\">\r\n<div id=\"fs-id1165135369594\">\r\n<p id=\"fs-id1165135369596\">Prove the conjecture made in the previous exercise.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135369601\">\r\n<div id=\"fs-id1165135369603\">\r\n<p id=\"fs-id1165135369606\">What is the domain of the function[latex]\\,f\\left(x\\right)=\\mathrm{ln}\\left(\\frac{x+2}{x-4}\\right)?\\,[\/latex]Discuss the result.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135593381\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135593381\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135593381\"]\r\n<p id=\"fs-id1165135593383\">Recall that the argument of a logarithmic function must be positive, so we determine where[latex]\\,\\frac{x+2}{x-4}&gt;0\\,[\/latex]. From the graph of the function[latex]\\,f\\left(x\\right)=\\frac{x+2}{x-4},[\/latex] note that the graph lies above the <em>x<\/em>-axis on the interval[latex]\\,\\left(-\\infty ,-2\\right)\\,[\/latex]and again to the right of the vertical asymptote, that is[latex]\\,\\left(4,\\infty \\right).\\,[\/latex]Therefore, the domain is[latex]\\,\\left(-\\infty ,-2\\right)\\cup \\left(4,\\infty \\right).[\/latex]<\/p>\r\n<span id=\"fs-id1165137838643\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140851\/CNX_Precalc_Figure_04_04_219.jpg\" alt=\"\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137838658\">\r\n<div id=\"fs-id1165137838660\">\r\n<p id=\"fs-id1165137838662\">Use properties of exponents to find the <em>x<\/em>-intercepts of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left({x}^{2}+4x+4\\right)\\,[\/latex]algebraically. Show the steps for solving, and then verify the result by graphing the function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\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.<\/li>\n<li>Graph logarithmic functions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135194555\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/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. In other words, logarithms give the <em>cause<\/em> for an <em>effect<\/em>.<\/p>\n<p id=\"fs-id1165137603580\">To illustrate, suppose we invest[latex]\\,\\text{\\$}2500\\,[\/latex]in an account that offers an annual interest rate of[latex]\\,5%,[\/latex]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 for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_001\">(Figure)<\/a> shows this point on the logarithmic graph.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140536\/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 \ud83d\udcb225,000 on the first year. The graph also notes that the balance reaches \ud83d\udcb25,000 near year 14.\" width=\"975\" height=\"497\" \/><\/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]\\,y={b}^{x}\\,[\/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]\\,y\\,[\/latex]is[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\n<li>The range of[latex]\\,y\\,[\/latex]is[latex]\\,\\left(0,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135641666\">In the last section we learned that the logarithmic function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is the inverse of the exponential function[latex]\\,y={b}^{x}.\\,[\/latex]So, as inverse functions:<\/p>\n<ul id=\"fs-id1165137656096\">\n<li>The domain of[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is the range of[latex]\\,y={b}^{x}:\\,[\/latex][latex]\\left(0,\\infty \\right).[\/latex]<\/li>\n<li>The range of[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is the domain of[latex]\\,y={b}^{x}:\\,[\/latex][latex]\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135245571\">Transformations of the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections\u2014to the parent function without loss of shape.<\/p>\n<p id=\"fs-id1165137653624\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/graphs-of-exponential-functions\/\">Graphs of Exponential Functions<\/a> we saw that certain transformations can change the <em>range<\/em> of[latex]\\,y={b}^{x}.\\,[\/latex]Similarly, applying transformations to the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]can change the <em>domain<\/em>. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists <em>only of positive real numbers<\/em>. That is, the argument of the logarithmic function must be greater than zero.<\/p>\n<p id=\"fs-id1165137851584\">For example, consider[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x-3\\right).\\,[\/latex]This function is defined for any values of[latex]\\,x\\,[\/latex]such that the argument, in this case[latex]\\,2x-3,[\/latex] is greater than zero. To find the domain, we set up an inequality and solve for[latex]\\,x:[\/latex]<\/p>\n<div id=\"eip-318\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}2x-3>0\\hfill & \\text{Show the argument greater than zero}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x>3\\hfill & \\text{Add 3}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>1.5\\begin{array}{cccc}& & & \\end{array}\\hfill & \\text{Divide by 2}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137645047\">In interval notation, the domain of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x-3\\right)\\,[\/latex]is[latex]\\,\\left(1.5,\\infty \\right).[\/latex]<\/p>\n<div id=\"fs-id1165137423048\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135173951\"><strong>Given a logarithmic function, identify the domain.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165137823224\" type=\"1\">\n<li>Set up an inequality showing the argument 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>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\" class=\"solution textbox shaded\">\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 is positive, so this function is defined when[latex]\\,x+3>0.\\,[\/latex]Solving this inequality,<\/p>\n<div id=\"eip-id1165135381135\" class=\"unnumbered\">[latex]\\begin{array}{ll}x+3>0\\hfill & \\text{The input must be positive}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>-3\\begin{array}{cccc}& & & \\end{array}\\hfill & \\text{Subtract 3}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137638183\">The domain of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)\\,[\/latex]is[latex]\\,\\left(-3,\\infty \\right).[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135692225\" class=\"textbox tryit\">\n<h3>Try It<\/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\" class=\"solution textbox shaded\">\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>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\" class=\"solution textbox shaded\">\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 is positive, so this function is defined when[latex]\\,5\u20132x>0.\\,[\/latex]Solving this inequality,<\/p>\n<div id=\"eip-id1165135470032\" class=\"unnumbered\">[latex]\\begin{array}{ll}5-2x>0\\hfill & \\text{The input must be positive}.\\hfill \\\\ \\,\\,\\,-2x>-5\\hfill & \\text{Subtract }5.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x<\\frac{5}{2}\\begin{array}{cccc}& & & \\end{array}\\hfill & \\text{Divide by }-2\\text{ and switch the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137656879\">The domain of[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(5-2x\\right)\\,[\/latex]is[latex]\\,\\left(\u2013\\infty ,\\frac{5}{2}\\right).[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874699\" class=\"textbox tryit\">\n<h3>Try It<\/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\" class=\"solution textbox shaded\">\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 parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]along with all its transformations: shifts, stretches, compressions, and reflections.<\/p>\n<p id=\"fs-id1165137679088\">We begin with the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right).\\,[\/latex]Because every logarithmic function of this form is the inverse of an exponential function with the form[latex]\\,y={b}^{x},[\/latex] their graphs will be reflections of each other across the line[latex]\\,y=x.\\,[\/latex]To illustrate this, we can observe the relationship between the input and output values of[latex]\\,y={2}^{x}\\,[\/latex]and its equivalent[latex]\\,x={\\mathrm{log}}_{2}\\left(y\\right)\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Table_04_04_01\">(Figure)<\/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>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135509175\">Using the inputs and outputs from <a class=\"autogenerated-content\" href=\"#Table_04_04_01\">(Figure)<\/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\">(Figure)<\/a><strong>.<\/strong><\/p>\n<table id=\"Table_04_04_02\" 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><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\n<td>[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\n<td>[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\n<td>[latex]\\left(0,1\\right)[\/latex]<\/td>\n<td>[latex]\\left(1,2\\right)[\/latex]<\/td>\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td>[latex]\\left(3,8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\n<td>[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\n<td>[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\n<td>[latex]\\left(1,0\\right)[\/latex]<\/td>\n<td>[latex]\\left(2,1\\right)[\/latex]<\/td>\n<td>[latex]\\left(4,2\\right)[\/latex]<\/td>\n<td>[latex]\\left(8,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137761335\">As we\u2019d expect, the <em>x<\/em>&#8211; and <em>y<\/em>-coordinates are reversed for the inverse functions. <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_002\">(Figure)<\/a> shows the graph of[latex]\\,f\\,[\/latex]and[latex]\\,g.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_002\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140543\/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. <\/strong>Notice that the graphs of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]and[latex]\\,g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)\\,[\/latex]are reflections about the line[latex]\\,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<div id=\"fs-id1165137780760\" class=\"textbox key-takeaways\">\n<h3>Characteristics of the Graph of the Parent Function, <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h3>\n<p id=\"fs-id1165135520250\">For any real number[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><em>x-<\/em>intercept:[latex]\\,\\left(1,0\\right)\\,[\/latex]and key point [latex]\\left(b,1\\right)[\/latex]<\/li>\n<li><em>y<\/em>-intercept: none<\/li>\n<li>increasing if[latex]\\,b>1[\/latex]<\/li>\n<li>decreasing if[latex]\\,0<b<1[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137464857\">See <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_003\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_003\" class=\"wp-caption aligncenter\">\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\/3252\/2018\/07\/19140545\/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<p id=\"fs-id1165137838178\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_004\">(Figure)<\/a> shows how changing the base[latex]\\,b\\,[\/latex]in[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (<em>Note:<\/em> recall that the function[latex]\\,\\mathrm{ln}\\left(x\\right)\\,[\/latex]has base[latex]\\,e\\approx \\text{2}.\\text{718.)}[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_004\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140548\/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 4. <\/strong>The graphs of three logarithmic functions with different bases, all greater than 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137430980\" class=\"bc-section section\">\n<div id=\"fs-id1165137871937\" class=\"precalculus howto textbox tryit\">\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 <em>x-<\/em>intercept,[latex]\\,\\left(1,0\\right).[\/latex]<\/li>\n<li>Plot the key point[latex]\\,\\left(b,1\\right).[\/latex]<\/li>\n<li>Draw a smooth curve through the points.<\/li>\n<li>State the domain,[latex]\\,\\left(0,\\infty \\right),[\/latex]the range,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]and the vertical asymptote,[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>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\" class=\"solution textbox shaded\">\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 tail of the graph will approach the vertical asymptote[latex]\\,x=0,[\/latex] and the right tail will increase slowly without bound.<\/li>\n<li>The <em>x<\/em>-intercept is[latex]\\,\\left(1,0\\right).[\/latex]<\/li>\n<li>The key point[latex]\\,\\left(5,1\\right)\\,[\/latex]is on the graph.<\/li>\n<li>We 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)<\/a>).<\/li>\n<\/ul>\n<div id=\"CNX_Precalc_Figure_04_04_005\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140555\/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=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/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 the vertical asymptote is[latex]\\,x=0.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135161300\" class=\"textbox tryit\">\n<h3>Try It<\/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 asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137407451\" class=\"solution textbox shaded\">\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 decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140603\/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.\" \/><\/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 the vertical asymptote is[latex]\\,x=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\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 other parent functions. We can shift, stretch, compress, and reflect the <span class=\"no-emphasis\">parent function<\/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 <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\n<p id=\"fs-id1165135530294\">When a constant[latex]\\,c\\,[\/latex]is added to the input of the parent function[latex]\\,f\\left(x\\right)=lo{g}_{b}\\left(x\\right),[\/latex] the result is a <span class=\"no-emphasis\">horizontal shift<\/span>[latex]\\,c\\,[\/latex]units in the <em>opposite<\/em> direction of the sign on[latex]\\,c.\\,[\/latex]To visualize horizontal shifts, we can observe the general graph of the parent function[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]and for[latex]\\,c>0\\,[\/latex]alongside the shift left,[latex]\\,g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right),[\/latex] and the shift right,[latex]\\,h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right).[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_04_04_007\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_04_04_007\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140605\/CNX_Precalc_Figure_04_04_007n.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.\" width=\"975\" height=\"570\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135296307\" class=\"textbox key-takeaways\">\n<h3>Horizontal Shifts of the Parent 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 parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]left[latex]\\,c\\,[\/latex]units if[latex]\\,c>0.[\/latex]<\/li>\n<li>shifts the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]right[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 id=\"fs-id1165137641710\" class=\"precalculus howto textbox tryit\">\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>Identify the horizontal shift:\n<ol id=\"fs-id1165137454288\" type=\"a\">\n<li>If[latex]\\,c>0,[\/latex]shift the graph of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]left[latex]\\,c\\,[\/latex]units.<\/li>\n<li>If[latex]\\,c<0,[\/latex]shift the graph of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]right[latex]\\,c\\,[\/latex]units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote[latex]\\,x=-c.[\/latex]<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting[latex]\\,c\\,[\/latex]from the[latex]\\,x\\,[\/latex]coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The Domain is[latex]\\,\\left(-c,\\infty \\right),[\/latex]the range is[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is[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>Graphing a Horizontal Shift of the Parent 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 its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137759883\" class=\"solution textbox shaded\">\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] we notice[latex]\\,x+\\left(-2\\right)=x\u20132.[\/latex]<\/p>\n<p id=\"fs-id1165137784630\">Thus[latex]\\,c=-2,[\/latex]so[latex]\\,c<0.\\,[\/latex]This means we will shift the function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\,[\/latex]right 2 units.<\/p>\n<p id=\"fs-id1165137836995\">The vertical asymptote is[latex]\\,x=-\\left(-2\\right)\\,[\/latex]or[latex]\\,x=2.[\/latex]<\/p>\n<p id=\"fs-id1165134042608\">Consider the three key points from the parent function,[latex]\\,\\left(\\frac{1}{3},-1\\right),[\/latex][latex]\\left(1,0\\right),[\/latex]and[latex]\\,\\left(3,1\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137475806\">The new coordinates are found by adding 2 to the[latex]\\,x\\,[\/latex]coordinates.<\/p>\n<p id=\"fs-id1165137748449\">Label the points[latex]\\,\\left(\\frac{7}{3},-1\\right),[\/latex][latex]\\left(3,0\\right),[\/latex]and[latex]\\,\\left(5,1\\right).[\/latex]<\/p>\n<p 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 aligncenter\">\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\/3252\/2018\/07\/19140608\/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=\"487\" height=\"363\" \/><\/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=\"textbox tryit\">\n<h3>Try It<\/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 its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165134373485\" class=\"solution textbox shaded\">\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 decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140610\/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).\" \/><\/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 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 parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right),[\/latex]the result is a <span class=\"no-emphasis\">vertical shift<\/span>[latex]\\,d\\,[\/latex]units in the direction of the sign on[latex]\\,d.\\,[\/latex]To visualize vertical shifts, we can observe the general graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]alongside the shift up,[latex]\\,g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\,[\/latex]and the shift down,[latex]\\,h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d.[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_04_04_010\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_04_04_010\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140612\/CNX_Precalc_Figure_04_04_010F.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.\" width=\"975\" height=\"741\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137767601\" class=\"textbox key-takeaways\">\n<h3>Vertical Shifts of the Parent 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 parent 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 parent 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 id=\"fs-id1165137706002\" class=\"precalculus howto textbox tryit\">\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]\\,f\\left(x\\right)={\\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]\\,f\\left(x\\right)={\\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 three key points from the parent function. Find new coordinates for the shifted functions by adding[latex]\\,d\\,[\/latex]to the[latex]\\,y\\,[\/latex]coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The domain is[latex]\\,\\left(0,\\infty \\right),[\/latex]the range is[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]and the vertical asymptote is[latex]\\,x=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_05\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137470059\">\n<h3>Graphing a Vertical Shift of the Parent 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 its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137925369\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137925369\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137925369\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137465913\">Since the function is[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2,[\/latex]we will notice[latex]\\,d=\u20132.\\,[\/latex]Thus[latex]\\,d<0.[\/latex]<\/p>\n<p id=\"fs-id1165135175015\">This means we will shift the function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\,[\/latex]down 2 units.<\/p>\n<p id=\"fs-id1165137644429\">The vertical asymptote is[latex]\\,x=0.[\/latex]<\/p>\n<p id=\"fs-id1165137408419\">Consider the three key points from the parent function,[latex]\\,\\left(\\frac{1}{3},-1\\right),[\/latex][latex]\\left(1,0\\right),[\/latex]and[latex]\\,\\left(3,1\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135503945\">The new coordinates are found by subtracting 2 from the <em>y <\/em>coordinates.<\/p>\n<p id=\"fs-id1165135421660\">Label the points[latex]\\,\\left(\\frac{1}{3},-3\\right),[\/latex][latex]\\left(1,-2\\right),[\/latex] and[latex]\\,\\left(3,-1\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135195524\">The domain is[latex]\\,\\left(0,\\infty \\right),[\/latex]the range is[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] and the vertical asymptote is[latex]\\,x=0.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_011\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140627\/CNX_Precalc_Figure_04_04_011.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=\"487\" height=\"516\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137698285\">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]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176716\" class=\"textbox tryit\">\n<h3>Try It<\/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 its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137874464\" class=\"small\">\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q676362\">Show Solution<\/span><\/p>\n<div id=\"q676362\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165137874471\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140633\/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).\" \/><\/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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137770245\" class=\"bc-section section\">\n<h4>Graphing Stretches and Compressions of <em>y<\/em> = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\n<p id=\"fs-id1165137418005\">When the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is multiplied by a constant[latex]\\,a>0,[\/latex] the result is a <span class=\"no-emphasis\">vertical stretch<\/span> or <span class=\"no-emphasis\">compression<\/span> of the original graph. To visualize stretches and compressions, we set[latex]\\,a>1\\,[\/latex]and observe the general graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]alongside the vertical stretch,[latex]\\,g\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]and the vertical compression,[latex]\\,h\\left(x\\right)=\\frac{1}{a}{\\mathrm{log}}_{b}\\left(x\\right).[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_04_04_013\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_04_04_013\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140638\/CNX_Precalc_Figure_04_04_013n.jpg\" alt=\"\" width=\"975\" height=\"758\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433996\" class=\"textbox key-takeaways\">\n<h3>Vertical Stretches and Compressions of the Parent 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 parent 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 parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]vertically by a factor of[latex]\\,a\\,[\/latex]if[latex]\\,0<a<1.[\/latex]<\/li>\n<li>has the vertical asymptote[latex]\\,x=0.[\/latex]<\/li>\n<li>has the <em>x<\/em>-intercept[latex]\\,\\left(1,0\\right).[\/latex]<\/li>\n<li>has domain[latex]\\,\\left(0,\\infty \\right).[\/latex]<\/li>\n<li>has range[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"precalculus howto\">\n<p id=\"fs-id1165135169307\"><strong>Given a logarithmic function with the form[latex]\\,f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right),[\/latex][latex]a>0,[\/latex]graph the translation.<\/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]\\,f\\left(x\\right)={\\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]\\,f\\left(x\\right)={\\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 three key points from the parent function. Find new coordinates for the shifted functions by multiplying the[latex]\\,y\\,[\/latex]coordinates by[latex]\\,a.[\/latex]<\/li>\n<li>Label the three points.<\/li>\n<li>The domain is[latex]\\,\\left(0,\\infty \\right),[\/latex]the range is[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]and the vertical asymptote is[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>Graphing a Stretch or Compression of the Parent 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 its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q420184\">Show Solution<\/span><\/p>\n<div id=\"q420184\" class=\"hidden-answer\" style=\"display: none\">Since the function is[latex]\\,f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right),[\/latex]we will notice[latex]\\,a=2.[\/latex]<\/p>\n<p id=\"fs-id1165135384321\">This means we will stretch the function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\,[\/latex]by a factor of 2.<\/p>\n<p id=\"fs-id1165135481989\">The vertical asymptote is[latex]\\,x=0.[\/latex]<\/p>\n<p id=\"fs-id1165137757801\">Consider the three key points from the parent function,[latex]\\,\\left(\\frac{1}{4},-1\\right),[\/latex][latex]\\left(1,0\\right),\\,[\/latex]and[latex]\\,\\left(4,1\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135570058\">The new coordinates are found by multiplying the[latex]\\,y\\,[\/latex]coordinates by 2.<\/p>\n<p id=\"fs-id1165137837989\">Label the points[latex]\\,\\left(\\frac{1}{4},-2\\right),[\/latex][latex]\\left(1,0\\right)\\,,[\/latex] and[latex]\\,\\left(4,\\text{2}\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135543469\">The domain is[latex]\\,\\left(0,\\,\\infty \\right),[\/latex] the range is[latex]\\,\\left(-\\infty ,\\infty \\right),\\,[\/latex]and the vertical asymptote is[latex]\\,x=0.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2455&amp;action=edit#CNX_Precalc_Figure_04_04_014\">(Figure)<\/a><strong>.<\/strong><\/p>\n<div id=\"CNX_Precalc_Figure_04_04_014\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140641\/CNX_Precalc_Figure_04_04_014.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135566827\">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>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134079655\" class=\"textbox tryit\">\n<h3>Try It<\/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}\\,{\\mathrm{log}}_{4}\\left(x\\right)\\,[\/latex]alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137778952\" class=\"solution textbox shaded\">\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 decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140643\/CNX_Precalc_Figure_04_04_015.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1\/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).\" \/><\/p>\n<p id=\"fs-id1165137768488\">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>\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>Combining a Shift and a Stretch<\/h3>\n<p id=\"fs-id1165137863045\">Sketch a graph of[latex]\\,f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\right).\\,[\/latex]State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137935559\" class=\"solution textbox shaded\">\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\">Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_016\">(Figure)<\/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 aligncenter\">\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\/3252\/2018\/07\/19140646\/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=\"487\" height=\"441\" \/><\/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]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560730\" class=\"textbox tryit\">\n<h3>Try It<\/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 asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137437228\" class=\"solution textbox shaded\">\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 decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140649\/CNX_Precalc_Figure_04_04_017.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" \/><\/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]<\/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 <em>f<\/em>(<em>x<\/em>) = log<sub><em>b<\/em><\/sub>(<em>x<\/em>)<\/h4>\n<p id=\"fs-id1165135169315\">When the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is multiplied by[latex]\\,-1,[\/latex]the result is a <span class=\"no-emphasis\">reflection<\/span> 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 parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]alongside the reflection about the <em>x<\/em>-axis,[latex]\\,g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)\\,[\/latex]and the reflection about the <em>y<\/em>-axis,[latex]\\,h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right).[\/latex]<\/p>\n<div id=\"Figure_04_04_018\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140655\/CNX_Precalc_Figure_04_04_018n.jpg\" alt=\"\" width=\"975\" height=\"786\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190744\" class=\"textbox key-takeaways\">\n<h3>Reflections of the Parent 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 id=\"fs-id1165137832285\">\n<li>reflects the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]about the <em>x<\/em>-axis.<\/li>\n<li>has domain,[latex]\\,\\left(0,\\infty \\right),[\/latex] range,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] and vertical asymptote,[latex]\\,x=0,[\/latex] which are unchanged from the parent 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 parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]about the <em>y<\/em>-axis.<\/li>\n<li>has domain[latex]\\,\\left(-\\infty ,0\\right).[\/latex]<\/li>\n<li>has range,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] and vertical asymptote,[latex]\\,x=0,[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"precalculus howto\">\n<p id=\"fs-id1165137638837\"><strong>Given a logarithmic function with the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] graph a translation.<\/strong><\/p>\n<table id=\"Table_04_04_08\" class=\"unnumbered\" summary=\"The first column gives the following instructions of graphing a translation of f(x)=-log_b(x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the x-axis; 4. Draw a smooth curve through the points; 5. State the domain, (0, infinity), the range, (-infinity, infinity), and the vertical asymptote x=0. The second column gives the following instructions of graphing a translation of f(x)=log_b(-x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (-1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the y-axis; 4. Draw a smooth curve through the points; 5. State the domain, (-infinity, 0), the range, (-infinity, infinity), and the vertical asymptote x=0.\">\n<thead>\n<tr>\n<th>[latex]\\text{If }f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\n<th>[latex]\\text{If }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\n<ol id=\"fs-id1165135313711\" type=\"1\">\n<li>Draw the vertical asymptote,[latex]\\,x=0.[\/latex]<\/li>\n<\/ol>\n<\/td>\n<td>\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>\n<td>\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>\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>\n<td>\n<ol id=\"fs-id1165135195408\" start=\"3\" type=\"1\">\n<li>Reflect the graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]about the <em>x<\/em>-axis.<\/li>\n<\/ol>\n<\/td>\n<td>\n<ol start=\"3\" type=\"1\">\n<li>Reflect the graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]about the <em>y<\/em>-axis.<\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<tr>\n<td>\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>\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>\n<td>\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>\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>Graphing a Reflection of a Logarithmic Function<\/h3>\n<p id=\"fs-id1165137849038\">Sketch a graph of[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)\\,[\/latex]alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137836523\" class=\"solution textbox shaded\">\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 parent function is increasing. Since the <em>input<\/em> value is multiplied by[latex]\\,-1,[\/latex][latex]f\\,[\/latex]is a reflection of the parent graph about the <em>y-<\/em>axis. Thus,[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)\\,[\/latex]will be decreasing as[latex]\\,x\\,[\/latex]moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote[latex]\\,x=0.\\,[\/latex]<\/li>\n<li>The <em>x<\/em>-intercept is[latex]\\,\\left(-1,0\\right).[\/latex]<\/li>\n<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\n<\/ul>\n<div id=\"CNX_Precalc_Figure_04_04_019\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140657\/CNX_Precalc_Figure_04_04_019.jpg\" alt=\"Graph of two functions. The parent function is y=log(x), with an asymptote at x=0 and labeled points at (1, 0), and (10, 0).The translation function f(x)=log(-x) has an asymptote at x=0 and labeled points at (-1, 0) and (-10, 1).\" width=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 14.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134042202\">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]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135650792\" class=\"textbox tryit\">\n<h3>Try It<\/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 asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137836698\" class=\"solution textbox shaded\">\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 decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140706\/CNX_Precalc_Figure_04_04_020.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" \/><\/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]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134579621\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134579627\"><strong>Given a logarithmic equation, use a graphing calculator to approximate solutions.<\/strong><\/p>\n<ol id=\"fs-id1165137431118\" type=\"1\">\n<li>Press <strong>[Y=]<\/strong>. Enter the given logarithm equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\n<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\n<li>To find the value of[latex]\\,x,[\/latex] we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \u201cintersect\u201d and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of[latex]\\,x,[\/latex]for the point(s) of intersection.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_09\" class=\"textbox examples\">\n<div id=\"fs-id1165135414229\">\n<div id=\"fs-id1165135414231\">\n<h3>Approximating the Solution of a Logarithmic Equation<\/h3>\n<p id=\"fs-id1165135414236\">Solve[latex]\\,4\\mathrm{ln}\\left(x\\right)+1=-2\\mathrm{ln}\\left(x-1\\right)\\,[\/latex]graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165135193432\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135193432\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135193432\" class=\"hidden-answer\" style=\"display: none\">Press <strong>[Y=]<\/strong> and enter[latex]\\,4\\mathrm{ln}\\left(x\\right)+1\\,[\/latex]next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter[latex]\\,-2\\mathrm{ln}\\left(x-1\\right)\\,[\/latex]next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for[latex]\\,x\\,[\/latex]and \u201310 to 10 for[latex]\\,y.\\,[\/latex]Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to right of[latex]\\,x=1.[\/latex]<\/p>\n<p id=\"fs-id1165135245763\">For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em>x<\/em>-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) So, to the nearest thousandth,[latex]\\,x\\approx 1.339.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137891344\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_04_09\">\n<div id=\"fs-id1165137639529\">\n<p id=\"fs-id1165137639531\">Solve[latex]\\,5\\mathrm{log}\\left(x+2\\right)=4-\\mathrm{log}\\left(x\\right)\\,[\/latex]graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165137817452\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137817452\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137817452\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137817454\">[latex]x\\approx 3.049[\/latex]<\/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\">(Figure)<\/a> to arrive at the general equation for translating exponential functions.<\/p>\n<table id=\"Table_04_04_009\">\n<caption>\u00a0<\/caption>\n<thead>\n<tr>\n<th colspan=\"2\">Translations of the Parent Function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<tr>\n<th>Translation<\/th>\n<th>Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift<\/p>\n<ul id=\"fs-id1165137416971\">\n<li>Horizontally[latex]\\,c\\,[\/latex]units to the left<\/li>\n<li>Vertically[latex]\\,d\\,[\/latex]units up<\/li>\n<\/ul>\n<\/td>\n<td>[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Stretch and Compress<\/p>\n<ul id=\"fs-id1165137427553\">\n<li>Stretch if[latex]\\,|a|>1[\/latex]<\/li>\n<li>Compression if[latex]\\,|a|<1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>x<\/em>-axis<\/td>\n<td>[latex]y=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>y<\/em>-axis<\/td>\n<td>[latex]y={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all translations<\/td>\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137414493\" class=\"textbox key-takeaways\">\n<h3>Translations of Logarithmic Functions<\/h3>\n<p id=\"fs-id1165137414501\">All translations of the parent logarithmic function,[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right),[\/latex] have the form<\/p>\n<div id=\"fs-id1165135408512\">[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/div>\n<p id=\"fs-id1165137734655\">where the parent function,[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right),b>1,[\/latex]is<\/p>\n<ul>\n<li>shifted vertically up[latex]\\,d\\,[\/latex]units.<\/li>\n<li>shifted horizontally to the left[latex]\\,c\\,[\/latex]units.<\/li>\n<li>stretched vertically by a factor of[latex]\\,|a|\\,[\/latex]if[latex]\\,|a|>0.[\/latex]<\/li>\n<li>compressed vertically by a factor of[latex]\\,|a|\\,[\/latex]if[latex]\\,0<|a|<1.[\/latex]<\/li>\n<li>reflected about the <em>x-<\/em>axis when[latex]\\,a<0.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137725084\">For[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(-x\\right),[\/latex] the graph of the parent function is reflected about the <em>y<\/em>-axis.<\/p>\n<\/div>\n<div id=\"Example_04_04_10\" class=\"textbox examples\">\n<div id=\"fs-id1165135296269\">\n<div id=\"fs-id1165135296271\">\n<h3>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\" class=\"solution textbox shaded\">\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 at[latex]\\,x=-4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137871955\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137871960\">The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to[latex]\\,x=-4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137748682\" class=\"textbox tryit\">\n<h3>Try It<\/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\" class=\"solution textbox shaded\">\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>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)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_021\" class=\"small aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140708\/CNX_Precalc_Figure_04_04_021.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 15.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135342977\" class=\"solution textbox shaded\">\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=\u20132\\,[\/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\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+k[\/latex]<\/div>\n<p id=\"fs-id1165135406913\">It appears the graph passes through the points[latex]\\,\\left(\u20131,1\\right)\\,[\/latex]and[latex]\\,\\left(2,\u20131\\right).\\,[\/latex]Substituting[latex]\\,\\left(\u20131,1\\right),[\/latex]<\/p>\n<div id=\"eip-id1165134101923\" class=\"unnumbered\">[latex]\\begin{array}{ll}1=-a\\mathrm{log}\\left(-1+2\\right)+k\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\hfill & \\text{Substitute }\\left(-1,1\\right).\\hfill \\\\ 1=-a\\mathrm{log}\\left(1\\right)+k\\hfill & \\text{Arithmetic}.\\hfill \\\\ 1=k\\hfill & \\text{log(1)}=0.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137628655\">Next, substituting in[latex]\\,\\left(2,\u20131\\right)[\/latex],<\/p>\n<div id=\"eip-id1165135431720\" class=\"unnumbered\">[latex]\\begin{array}{lll}-1=-a\\mathrm{log}\\left(2+2\\right)+1\\hfill & \\hfill & \\text{Plug in }\\left(2,-1\\right).\\hfill \\\\ -2=-a\\mathrm{log}\\left(4\\right)\\hfill & \\hfill & \\text{Arithmetic}.\\hfill \\\\ \\text{ }a=\\frac{2}{\\mathrm{log}\\left(4\\right)}\\hfill & \\hfill & \\text{Solve for }a.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135192211\">This gives us the equation[latex]\\,f\\left(x\\right)=\u2013\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735581\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137735586\">We can verify this answer by comparing the function values in <a class=\"autogenerated-content\" href=\"#Table_04_04_010\">(Figure)<\/a> with the points on the graph in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_021\">(Figure)<\/a>.<\/p>\n<table id=\"Table_04_04_010\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>\u22120.58496<\/td>\n<td>\u22121<\/td>\n<td>\u22121.3219<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>\u22121.5850<\/td>\n<td>\u22121.8074<\/td>\n<td>\u22122<\/td>\n<td>\u22122.1699<\/td>\n<td>\u22122.3219<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1525883\" class=\"textbox tryit\">\n<h3>Try It<\/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)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_022\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140711\/CNX_Precalc_Figure_04_04_022.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 16.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436435\" class=\"solution textbox shaded\">\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 textbox shaded\">\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)<\/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\\,|\\,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:\/\/openstaxcollege.org\/l\/graphexplog\">Graph an Exponential Function and Logarithmic Function<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/matchexplog\">Match Graphs with Exponential and Logarithmic Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.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>General Form for the Translation of the Parent Logarithmic Function[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/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 argument greater than zero, and solve for[latex]\\,x.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_04_01\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_04_04_02\">(Figure)<\/a><\/li>\n<li>The graph of the parent function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]has an <em>x-<\/em>intercept at[latex]\\,\\left(1,0\\right),[\/latex]domain[latex]\\,\\left(0,\\infty \\right),[\/latex]range[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]vertical asymptote[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<b<1,[\/latex] the function is decreasing.<\/li>\n<\/ul>\n<p>See <a class=\"autogenerated-content\" href=\"#Example_04_04_03\">(Figure)<\/a>.<\/li>\n<li>The equation[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)\\,[\/latex]shifts the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]horizontally\n<ul id=\"fs-id1165135512562\">\n<li>left[latex]\\,c\\,[\/latex]units if[latex]\\,c>0.[\/latex]<\/li>\n<li>right[latex]\\,c\\,[\/latex]units if[latex]\\,c<0.[\/latex]<\/li>\n<\/ul>\n<p>See <a class=\"autogenerated-content\" href=\"#Example_04_04_04\">(Figure)<\/a>.<\/li>\n<li>The equation[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\,[\/latex]shifts the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]vertically\n<ul id=\"fs-id1165137761068\">\n<li>up[latex]\\,d\\,[\/latex]units if[latex]\\,d>0.[\/latex]<\/li>\n<li>down[latex]\\,d\\,[\/latex]units if[latex]\\,d<0.[\/latex]<\/li>\n<\/ul>\n<p>See <a class=\"autogenerated-content\" href=\"#Example_04_04_05\">(Figure)<\/a>.<\/li>\n<li>For any constant[latex]\\,a>0,[\/latex] the equation[latex]\\,f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\n<ul id=\"fs-id1165134040579\">\n<li>stretches the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]vertically by a factor of[latex]\\,a\\,[\/latex]if[latex]\\,|a|>1.[\/latex]<\/li>\n<li>compresses the parent 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<p>See <a class=\"autogenerated-content\" href=\"#Example_04_04_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_04_04_07\">(Figure)<\/a>.<\/li>\n<li>When the parent function[latex]\\,y={\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]is multiplied by[latex]\\,-1,[\/latex] the result is a reflection about the <em>x<\/em>-axis. When the input is multiplied by[latex]\\,-1,[\/latex] the result is a reflection about the <em>y<\/em>-axis.\n<ul id=\"fs-id1165135186594\">\n<li>The equation[latex]\\,f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)\\,[\/latex]represents a reflection of the parent function about the <em>x-<\/em>axis.<\/li>\n<li>The equation[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)\\,[\/latex]represents a reflection of the parent function about the <em>y-<\/em>axis.<\/li>\n<\/ul>\n<p>See <a class=\"autogenerated-content\" href=\"#Example_04_04_08\">(Figure)<\/a>.<\/p>\n<ul id=\"fs-id1165137834414\">\n<li>A graphing calculator may be used to approximate solutions to some logarithmic equations See <a class=\"autogenerated-content\" href=\"#Example_04_04_09\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/li>\n<li>All translations of the logarithmic function can be summarized by the general equation[latex]\\, f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Table_04_04_009\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_04_04_10\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_04_04_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135189917\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135189921\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135582182\">\n<div id=\"fs-id1165135582184\">\n<p id=\"fs-id1165135582187\">The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?<\/p>\n<\/div>\n<div id=\"fs-id1165135582193\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135582193\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135582193\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135582195\">Since the functions are inverses, their graphs are mirror images about the line[latex]\\,y=x.\\,[\/latex]So for every point[latex]\\,\\left(a,b\\right)\\,[\/latex]on the graph of a logarithmic function, there is a corresponding point[latex]\\,\\left(b,a\\right)\\,[\/latex]on the graph of its inverse exponential function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137855282\">\n<div id=\"fs-id1165137855284\">\n<p id=\"fs-id1165137855286\">What type(s) of translation(s), if any, affect the range of a logarithmic function?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137855292\">\n<div id=\"fs-id1165137855294\">\n<p id=\"fs-id1165137855296\">What type(s) of translation(s), if any, affect the domain of a logarithmic function?<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q917821\">Show Solution<\/span><\/p>\n<div id=\"q917821\" class=\"hidden-answer\" style=\"display: none\">Shifting the function right or left and reflecting the function about the y-axis will affect its domain.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137424694\">\n<p id=\"fs-id1165137424697\">Consider the general logarithmic function[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right).\\,[\/latex]Why can\u2019t[latex]\\,x\\,[\/latex]be zero?<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137697130\">Does the graph of a general logarithmic function have a horizontal asymptote? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1165137459912\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137459912\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137459912\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137459914\">No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137459922\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137459927\">For the following exercises, state the domain and range of the function.<\/p>\n<div id=\"fs-id1165137459930\">\n<div id=\"fs-id1165137737591\">[latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135264771\">\n<div id=\"fs-id1165135264773\">\n<p id=\"fs-id1165135264775\">[latex]h\\left(x\\right)=\\mathrm{ln}\\left(\\frac{1}{2}-x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135257219\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135257219\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135257219\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135257221\">Domain:[latex]\\,\\left(-\\infty ,\\frac{1}{2}\\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135253806\">\n<div id=\"fs-id1165135253808\">\n<p id=\"fs-id1165135253810\">[latex]g\\left(x\\right)={\\mathrm{log}}_{5}\\left(2x+9\\right)-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135415813\">\n<div id=\"fs-id1165135415815\">\n<p id=\"fs-id1165135415817\">[latex]h\\left(x\\right)=\\mathrm{ln}\\left(4x+17\\right)-5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137641046\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137641046\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137641046\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137641048\">Domain:[latex]\\,\\left(-\\frac{17}{4},\\infty \\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137843712\">\n<div id=\"fs-id1165137843714\">\n<p id=\"fs-id1165137843716\">[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(12-3x\\right)-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135503696\">For the following exercises, state the domain and the vertical asymptote of the function.<\/p>\n<div id=\"fs-id1165134037565\">\n<div id=\"fs-id1165134037567\">\n<p id=\"fs-id1165134037569\">[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-5\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137771436\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137771436\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137771436\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137771438\">Domain:[latex]\\,\\left(5,\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137933908\">\n<div id=\"fs-id1165137933910\">\n<p id=\"fs-id1165137933912\">[latex]\\,g\\left(x\\right)=\\mathrm{ln}\\left(3-x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737027\">\n<div id=\"fs-id1165137737029\">\n<p id=\"fs-id1165137737031\">[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left(3x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135256140\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135256140\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135256140\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135256142\">Domain:[latex]\\,\\left(-\\frac{1}{3},\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=-\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188081\">\n<div id=\"fs-id1165135188083\">\n<p id=\"fs-id1165135188086\">[latex]\\,f\\left(x\\right)=3\\mathrm{log}\\left(-x\\right)+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134188310\">\n<div id=\"fs-id1165134188313\">\n<p id=\"fs-id1165134188315\">[latex]\\,g\\left(x\\right)=-\\mathrm{ln}\\left(3x+9\\right)-7[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135532384\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135532384\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135532384\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135208715\">Domain:[latex]\\,\\left(-3,\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135194730\">For the following exercises, state the domain, vertical asymptote, and end behavior of the function.<\/p>\n<div id=\"fs-id1165135194734\">\n<div id=\"fs-id1165135194736\">\n<p id=\"fs-id1165135194738\">[latex]f\\left(x\\right)=\\mathrm{ln}\\left(2-x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137427662\">\n<div id=\"fs-id1165137427665\">\n<p id=\"fs-id1165137427667\">[latex]f\\left(x\\right)=\\mathrm{log}\\left(x-\\frac{3}{7}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137715430\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137715430\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137715430\" class=\"hidden-answer\" style=\"display: none\">Domain: [latex]\\left(\\frac{3}{7},\\infty \\right)[\/latex];<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137431240\">\n<div id=\"fs-id1165137431242\">\n<p id=\"fs-id1165137431244\">[latex]h\\left(x\\right)=-\\mathrm{log}\\left(3x-4\\right)+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134194950\">\n<div id=\"fs-id1165134194952\">\n<p id=\"fs-id1165134194954\">[latex]g\\left(x\\right)=\\mathrm{ln}\\left(2x+6\\right)-5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135358044\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135358044\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135358044\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain: [latex]\\left(-3,\\infty \\right)[\/latex]; Vertical asymptote: [latex]x=-3[\/latex];<\/p><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135532457\">\n<div id=\"fs-id1165135532459\">\n<p id=\"fs-id1165135532461\">[latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(15-5x\\right)+6[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137426354\">For the following exercises, state the domain, range, and <em>x<\/em>&#8211; and <em>y<\/em>-intercepts, if they exist. If they do not exist, write DNE.<\/p>\n<div id=\"fs-id1165137426368\">\n<div id=\"fs-id1165137426370\">\n<p id=\"fs-id1165137426372\">[latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x-1\\right)+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132983045\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165132983045\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165132983045\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165132983048\">Domain:[latex]\\,\\left(1,\\infty \\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=1;\\,[\/latex]<em>x<\/em>-intercept:[latex]\\,\\left(\\frac{5}{4},0\\right);\\,[\/latex]<em>y<\/em>-intercept: DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137755640\">\n<div id=\"fs-id1165135543374\">\n<p id=\"fs-id1165135543376\">[latex]f\\left(x\\right)=\\mathrm{log}\\left(5x+10\\right)+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135210139\">\n<div id=\"fs-id1165135210141\">\n<p id=\"fs-id1165135210143\">[latex]g\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)-2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137426893\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137426893\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137426893\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137426895\">Domain:[latex]\\,\\left(-\\infty ,0\\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]Vertical asymptote:[latex]\\,x=0;\\,[\/latex]<em>x<\/em>-intercept:[latex]\\,\\left(-{e}^{2},0\\right);\\,[\/latex]<em>y<\/em>-intercept: DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810010\">\n<div id=\"fs-id1165137810012\">\n<p id=\"fs-id1165135401817\">[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)-5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137942453\">\n<div id=\"fs-id1165137942455\">[latex]h\\left(x\\right)=3\\mathrm{ln}\\left(x\\right)-9[\/latex]<\/div>\n<div id=\"fs-id1165135186502\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135186502\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135186502\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135186504\">Domain:[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]Range:[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex] Vertical asymptote: [latex]\\,x=0;\\,[\/latex]<em>x<\/em>-intercept:[latex]\\,\\left({e}^{3},0\\right);\\,[\/latex]<em>y<\/em>-intercept: DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137422264\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137422269\">For the following exercises, match each function in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_201\">(Figure)<\/a> with the letter corresponding to its graph.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_201\" class=\"small aligncenter\">\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\/3252\/2018\/07\/19140718\/CNX_PreCalc_Figure_04_04_201.jpg\" alt=\"Graph of five logarithmic functions.\" width=\"487\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 17.<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134032273\">\n<div id=\"fs-id1165134032275\">\n<p id=\"fs-id1165134032277\">[latex]d\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135194197\">\n<div id=\"fs-id1165137911558\">\n<p id=\"fs-id1165137911560\">[latex]f\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134255558\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134255558\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134255558\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134255561\">B<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134255566\">\n<div id=\"fs-id1165134255568\">\n<p id=\"fs-id1165134255570\">[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758154\">\n<div id=\"fs-id1165137758156\">\n<p id=\"fs-id1165137758158\">[latex]h\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135571660\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135571660\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135571660\">C<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571665\">\n<div>\n<p id=\"fs-id1165135571669\">[latex]j\\left(x\\right)={\\mathrm{log}}_{25}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>For the following exercises, match each function in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_202\">(Figure)<\/a> with the letter corresponding to its graph.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_202\" class=\"small aligncenter\">\n<div style=\"width: 352px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140728\/CNX_PreCalc_Figure_04_04_202.jpg\" alt=\"Graph of three logarithmic functions.\" width=\"342\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 18.<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134380389\">\n<div id=\"fs-id1165135400154\">\n<p id=\"fs-id1165135400156\">[latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{3}}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135206082\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135206082\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135206082\" class=\"hidden-answer\" style=\"display: none\">B<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135206089\">\n<div id=\"fs-id1165135206092\">\n<p id=\"fs-id1165135206094\">[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137855381\">\n<div id=\"fs-id1165137855383\">\n<p id=\"fs-id1165137855385\">[latex]h\\left(x\\right)={\\mathrm{log}}_{\\frac{3}{4}}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137637948\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137637948\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137637948\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135394336\">C<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-967\">For the following exercises, sketch the graphs of each pair of functions on the same axis.<\/p>\n<div id=\"fs-id1165135394346\">\n<div id=\"fs-id1165135394348\">\n<p id=\"fs-id1165135394350\">[latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)={10}^{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137862457\">\n<div id=\"fs-id1165137862459\">\n<p id=\"fs-id1165137862461\">[latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q809684\">Show Solution<\/span><\/p>\n<div id=\"q809684\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140735\/CNX_PreCalc_Figure_04_04_204.jpg\" alt=\"Graph of two functions, g(x) = log_(1\/2)(x) in orange and f(x)=log(x) in blue.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135180056\">\n<div id=\"fs-id1165135180058\">\n<p id=\"fs-id1165135180060\">[latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135536589\">\n<div id=\"fs-id1165134325846\">\n<p id=\"fs-id1165134325848\">[latex]f\\left(x\\right)={e}^{x}\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135496605\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135496605\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135496605\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165135496611\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140747\/CNX_PreCalc_Figure_04_04_206.jpg\" alt=\"Graph of two functions, g(x) = ln(1\/2)(x) in orange and f(x)=e^(x) in blue.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135496626\">For the following exercises, match each function in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_04_207\">(Figure)<\/a> with the letter corresponding to its graph.<\/p>\n<div id=\"CNX_Precalc_Figure_04_04_207\" class=\"small aligncenter\">\n<div style=\"width: 384px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140759\/CNX_Precalc_Figure_04_04_207.jpg\" alt=\"Graph of three logarithmic functions.\" width=\"374\" height=\"377\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 19.<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189876\">\n<div id=\"fs-id1165135189878\">\n<p id=\"fs-id1165135189880\">[latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(-x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656835\">\n<div id=\"fs-id1165137656837\">\n<p id=\"fs-id1165137656839\">[latex]g\\left(x\\right)=-{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div id=\"fs-id1165137656835\">\n<div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q478955\">Show Solution<\/span><\/p>\n<div id=\"q478955\" class=\"hidden-answer\" style=\"display: none\">C<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135632110\">\n<div id=\"fs-id1165135632112\">\n<p id=\"fs-id1165135408522\">[latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>For the following exercises, sketch the graph of the indicated function.<\/p>\n<div id=\"fs-id1165135571798\">\n<div id=\"fs-id1165135571800\">\n<p id=\"fs-id1165135571802\">[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"eip-id2073097\">\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q916976\">Show Solution<\/span><\/p>\n<div id=\"q916976\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140801\/CNX_PreCalc_Figure_04_04_208.jpg\" alt=\"Graph of f(x)=log_2(x+2).\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137849072\">\n<div id=\"fs-id1165137849074\">\n<p id=\"fs-id1165137849076\">[latex]\\,f\\left(x\\right)=2\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251375\">\n<div id=\"fs-id1165135251377\">\n<p id=\"fs-id1165135251380\">[latex]\\,f\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134032402\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134032402\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134032402\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165134032408\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140809\/CNX_PreCalc_Figure_04_04_210.jpg\" alt=\"Graph of f(x)=ln(-x).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134032422\">\n<div id=\"fs-id1165134032425\">\n<p id=\"fs-id1165134032427\">[latex]g\\left(x\\right)=\\mathrm{log}\\left(4x+16\\right)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932668\">\n<div id=\"fs-id1165137932670\">\n<p id=\"fs-id1165137932672\">[latex]g\\left(x\\right)=\\mathrm{log}\\left(6-3x\\right)+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137705397\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137705397\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137705397\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140814\/CNX_PreCalc_Figure_04_04_212.jpg\" alt=\"Graph of g(x)=log(6-3x)+1.\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705417\">\n<div id=\"fs-id1165137705419\">\n<p id=\"fs-id1165137563331\">[latex]h\\left(x\\right)=-\\frac{1}{2}\\mathrm{ln}\\left(x+1\\right)-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135439867\">For the following exercises, write a logarithmic equation corresponding to the graph shown.<\/p>\n<div id=\"fs-id1165135439871\">\n<div id=\"fs-id1165135439873\">\n<p id=\"fs-id1165135439875\">Use[latex]\\,y={\\mathrm{log}}_{2}\\left(x\\right)\\,[\/latex]as the parent function.<\/p>\n<p><span id=\"fs-id1165135443953\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140818\/CNX_PreCalc_Figure_04_04_214.jpg\" alt=\"The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165134360851\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134360851\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134360851\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134360853\">[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{2}\\left(-\\left(x-1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191066\">\n<div id=\"fs-id1165135191068\">\n<p id=\"fs-id1165135191070\">Use[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\,[\/latex]as the parent function.<\/p>\n<p><span id=\"fs-id1165137674237\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140830\/CNX_PreCalc_Figure_04_04_215.jpg\" alt=\"The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137674252\">\n<div id=\"fs-id1165137674255\">\n<p id=\"fs-id1165137674257\">Use[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\,[\/latex]as the parent function.<\/p>\n<p><span id=\"fs-id1165134086070\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140833\/CNX_PreCalc_Figure_04_04_216.jpg\" alt=\"The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165134086084\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134086084\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134086084\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134086086\">[latex]f\\left(x\\right)=3{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203463\">\n<div id=\"fs-id1165135203465\">\n<p id=\"fs-id1165135203467\">Use[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)\\,[\/latex]as the parent function.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140840\/CNX_PreCalc_Figure_04_04_217.jpg\" alt=\"The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705205\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137705211\">For the following exercises, use a graphing calculator to find approximate solutions to each equation.<\/p>\n<div id=\"fs-id1165137705215\">\n<div>\n<p id=\"fs-id1165135547452\">[latex]\\mathrm{log}\\left(x-1\\right)+2=\\mathrm{ln}\\left(x-1\\right)+2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137674291\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137674291\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137674291\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137674294\">[latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137674311\">\n<div id=\"fs-id1165137674313\">\n<p id=\"fs-id1165135438432\">[latex]\\mathrm{log}\\left(2x-3\\right)+2=-\\mathrm{log}\\left(2x-3\\right)+5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135176229\">\n<p id=\"fs-id1165135176231\">[latex]\\mathrm{ln}\\left(x-2\\right)=-\\mathrm{ln}\\left(x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135337765\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135337765\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135337765\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135181772\">[latex]x\\approx \\text{2}\\text{.303}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181791\">\n<div id=\"fs-id1165135181793\">\n<p id=\"fs-id1165135181795\">[latex]2\\mathrm{ln}\\left(5x+1\\right)=\\frac{1}{2}\\mathrm{ln}\\left(-5x\\right)+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135367864\">\n<div id=\"fs-id1165135367866\">\n<p id=\"fs-id1165135367869\">[latex]\\frac{1}{3}\\mathrm{log}\\left(1-x\\right)=\\mathrm{log}\\left(x+1\\right)+\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135174505\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135174505\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135174505\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135485716\">[latex]x\\approx -0.472[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135485736\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div>\n<div id=\"fs-id1165135485744\">\n<p id=\"fs-id1165135485746\">Let[latex]\\,b\\,[\/latex]be any positive real number such that[latex]\\,b\\ne 1.\\,[\/latex]What must[latex]\\,{\\mathrm{log}}_{b}1\\,[\/latex]be equal to? Verify the result.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193274\">\n<div id=\"fs-id1165135193276\">\n<p id=\"fs-id1165135193279\">Explore and discuss the graphs of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right).\\,[\/latex]Make a conjecture based on the result.<\/p>\n<\/div>\n<div id=\"fs-id1165134151930\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134151930\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134151930\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134151932\">The graphs of[latex]\\,f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right)\\,[\/latex]appear to be the same; Conjecture: for any positive base[latex]\\,b\\ne 1,[\/latex][latex]\\,{\\mathrm{log}}_{b}\\left(x\\right)=-{\\mathrm{log}}_{\\frac{1}{b}}\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135369592\">\n<div id=\"fs-id1165135369594\">\n<p id=\"fs-id1165135369596\">Prove the conjecture made in the previous exercise.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135369601\">\n<div id=\"fs-id1165135369603\">\n<p id=\"fs-id1165135369606\">What is the domain of the function[latex]\\,f\\left(x\\right)=\\mathrm{ln}\\left(\\frac{x+2}{x-4}\\right)?\\,[\/latex]Discuss the result.<\/p>\n<\/div>\n<div id=\"fs-id1165135593381\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135593381\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135593381\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135593383\">Recall that the argument of a logarithmic function must be positive, so we determine where[latex]\\,\\frac{x+2}{x-4}>0\\,[\/latex]. From the graph of the function[latex]\\,f\\left(x\\right)=\\frac{x+2}{x-4},[\/latex] note that the graph lies above the <em>x<\/em>-axis on the interval[latex]\\,\\left(-\\infty ,-2\\right)\\,[\/latex]and again to the right of the vertical asymptote, that is[latex]\\,\\left(4,\\infty \\right).\\,[\/latex]Therefore, the domain is[latex]\\,\\left(-\\infty ,-2\\right)\\cup \\left(4,\\infty \\right).[\/latex]<\/p>\n<p><span id=\"fs-id1165137838643\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140851\/CNX_Precalc_Figure_04_04_219.jpg\" alt=\"\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838658\">\n<div id=\"fs-id1165137838660\">\n<p id=\"fs-id1165137838662\">Use properties of exponents to find the <em>x<\/em>-intercepts of the function[latex]\\,f\\left(x\\right)=\\mathrm{log}\\left({x}^{2}+4x+4\\right)\\,[\/latex]algebraically. Show the steps for solving, and then verify the result by graphing the function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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-2455\">\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>Algebra and Trigonometry. <strong>Authored by<\/strong>: Jay Abramson, et. al. <strong>Provided by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\">http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/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":53384,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Algebra and Trigonometry\",\"author\":\"Jay Abramson, et. al\",\"organization\":\"OpenStax CNX\",\"url\":\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2455","chapter","type-chapter","status-publish","hentry"],"part":2358,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2455","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/users\/53384"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2455\/revisions"}],"predecessor-version":[{"id":3728,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2455\/revisions\/3728"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/parts\/2358"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2455\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/media?parent=2455"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapter-type?post=2455"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/contributor?post=2455"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/license?post=2455"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}