{"id":273,"date":"2023-06-21T13:22:51","date_gmt":"2023-06-21T13:22:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/stretch-compress-or-reflect-a-logarithmic-function\/"},"modified":"2023-07-04T04:19:52","modified_gmt":"2023-07-04T04:19:52","slug":"stretch-compress-or-reflect-a-logarithmic-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/stretch-compress-or-reflect-a-logarithmic-function\/","title":{"raw":"\u25aa   Stretching, Compressing, or Reflecting a Logarithmic Function","rendered":"\u25aa   Stretching, Compressing, or Reflecting a Logarithmic Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph stretches and compressions of logarithmic functions.<\/li>\r\n \t<li>Graph reflections of logarithmic functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing Stretches and Compressions of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h2>\r\nWhen the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by a constant <em>a<\/em> &gt; 0, the result is a <strong>vertical stretch<\/strong> or <strong>compression<\/strong> of the original graph. To visualize stretches and compressions, we set <em>a\u00a0<\/em>&gt; 1 and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the vertical stretch, [latex]g\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], and the vertical compression, [latex]h\\left(x\\right)=\\frac{1}{a}{\\mathrm{log}}_{b}\\left(x\\right)[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nUsing an online graphing calculator plot the functions [latex]g(x) = a\\log_{b}{x}[\/latex]\u00a0 and\u00a0 [latex]h(x) = \\frac{1}{a}\\log_{b}{x}[\/latex]. One represents a vertical compression of the other. You may select any value for [latex]b[\/latex], though one between [latex]2[\/latex] and [latex]5[\/latex] will be easier to see. Experiment with various [latex]a[\/latex] values between [latex]1[\/latex] and [latex]10[\/latex]. As you investigate, consider the following questions:\r\n<ul>\r\n \t<li>Both the vertical stretch and compression produce graphs that are increasing. Which transformation produces a function that increases faster?<\/li>\r\n \t<li>One of the key points that is commonly defined for transformations of a logarithmic function comes from finding the input that gives an output of [latex]y = 1[\/latex]. This point can help you determine whether a graph is the result of a vertical compression or stretch. Explain why.<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe graphs below summarize the key features of the resulting graphs of vertical stretches and compressions of logarithmic functions.\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233836\/CNX_Precalc_Figure_04_04_013n2.jpg\" alt=\"&quot;Graph\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Vertical Stretches and Compressions of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nFor any constant <em>a<\/em> &gt; 1, the function [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\r\n<ul>\r\n \t<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if <em>a\u00a0<\/em>&gt; 1.<\/li>\r\n \t<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if 0 &lt; <em>a\u00a0<\/em>&lt; 1.<\/li>\r\n \t<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>has the <em>x<\/em>-intercept [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n \t<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a logarithmic function Of the form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], [latex]a&gt;0[\/latex], graph the Stretch or Compression<\/h3>\r\n<ol>\r\n \t<li>Identify the vertical stretch or compression:\r\n<ul>\r\n \t<li>If [latex]|a|&gt;1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is stretched by a factor of <em>a<\/em>\u00a0units.<\/li>\r\n \t<li>If [latex]|a|&lt;1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is compressed by a factor of <em>a<\/em>\u00a0units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the <em>y<\/em>\u00a0coordinates in each point by <em>a<\/em>.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x<\/em> = 0.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nLogarithm functions follow the same principles as the other toolkit functions with regard to stretches, compressions, and reflections.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Stretch or Compression of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nSketch the 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.\r\n\r\n[reveal-answer q=\"595868\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"595868\"]\r\n\r\nSince the function is [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex], we will note that<i> <\/i>= 2.\r\n\r\nThis means we will stretch the function [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] by a factor of 2.\r\n\r\nThe vertical asymptote is <em>x\u00a0<\/em>= 0.\r\n\r\nConsider 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].\r\n\r\nThe new coordinates are found by multiplying the <em>y<\/em>\u00a0coordinates of each point by 2.\r\n\r\nLabel the points [latex]\\left(\\frac{1}{4},-2\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(4,\\text{2}\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233838\/CNX_Precalc_Figure_04_04_0142.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).\" width=\"487\" height=\"366\" \/> The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x\u00a0= 0.[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSketch 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.\r\n\r\n[reveal-answer q=\"250125\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"250125\"]\r\n\r\nThe domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.\r\n\r\n<img class=\"aligncenter size-full wp-image-3114\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16233042\/CNX_Precalc_Figure_04_04_0152.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1\/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).\" width=\"487\" height=\"364\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Combining a Shift and a Stretch<\/h3>\r\nSketch the graph of [latex]f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\right)[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"804029\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"804029\"]\r\n\r\nRemember, what happens inside parentheses happens first. First, we move the graph left 2 units and then stretch the function vertically by a factor of 5. The vertical asymptote will be shifted to <em>x\u00a0<\/em>= \u20132. The <em>x<\/em>-intercept will be [latex]\\left(-1,0\\right)[\/latex]. The domain will be [latex]\\left(-2,\\infty \\right)[\/latex]. Two points will help give the shape of the graph: [latex]\\left(-1,0\\right)[\/latex] and [latex]\\left(8,5\\right)[\/latex]. We chose <em>x\u00a0<\/em>= 8 as the <em>x<\/em>-coordinate of one point to graph because when <em>x\u00a0<\/em>= 8, <em>x\u00a0<\/em>+ 2 = 10, the base of the common logarithm.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233840\/CNX_Precalc_Figure_04_04_0162.jpg\" alt=\"Graph of three functions. The parent function is y=log(x), with an asymptote at x=0. The first translation function y=5log(x+2) has an asymptote at x=-2. The second translation function y=log(x+2) has an asymptote at x=-2.\" width=\"487\" height=\"441\" \/> The domain is [latex]\\left(-2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x\u00a0= \u20132.[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSketch 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.\r\n\r\n[reveal-answer q=\"404704\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"404704\"]\r\n\r\nThe domain is [latex]\\left(2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.\r\n\r\n<img class=\"aligncenter size-full wp-image-3115\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16233717\/CNX_Precalc_Figure_04_04_0172.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" width=\"487\" height=\"439\" \/>\r\n<div id=\"fs-id1165137437228\" class=\"solution\"><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Graphing Reflections of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h2>\r\nWhen the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a <strong>reflection<\/strong> about the <em>x<\/em>-axis. When the <em>input<\/em> is multiplied by \u20131, the result is a reflection about the <em>y<\/em>-axis. To visualize reflections, we restrict <em>b\u00a0<\/em>&gt; 1 and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex], and the reflection about the <em>y<\/em>-axis, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nUsing an online graphing calculator, plot the functions [latex]f(x) = \\log_{b}{x},\\text{ }g(x)=-\\log_{b}{x},\\text{ and }h(x) = \\log_{b}({-x}) [\/latex]. You may select any value for [latex]b[\/latex], though one between [latex]2[\/latex] and [latex]5[\/latex] will be easier to see. Also plot the point [latex](b,1)[\/latex]. Consider the following questions:\r\n<ul>\r\n \t<li>Which graph, [latex]g(x) = -\\log_{b}{x} \\text{ or }h(x) = \\log_{b}({-x})[\/latex] represents a vertical reflection? \u00a0Which one represents a horizontal reflection?<\/li>\r\n \t<li>You already added the point [latex](b,1)[\/latex] as a point of interest for the function [latex]f(x)[\/latex]. Using the variable [latex]b[\/latex] as your [latex]x[\/latex] value, add the corresponding points of interest for [latex]g(x)\\text{ and }h(x)[\/latex].<\/li>\r\n \t<li>Does the vertical asymptote change when you reflect the graph of [latex]f(x)[\/latex] either vertically or horizontally?<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe graphs below summarize the key characteristics of reflecting [latex]f(x) = \\log_{b}{x}[\/latex] horizontally and vertically.\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233843\/CNX_Precalc_Figure_04_04_018n2.jpg\" alt=\"&quot;Graph\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Reflections of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nThe function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex]\r\n<ul>\r\n \t<li>reflects the 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\u00a0<em>x\u00a0<\/em>= 0 which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\nThe function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]\r\n<ul>\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\u00a0<em>x\u00a0<\/em>= 0 which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a logarithmic function with the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph a Reflection<\/h3>\r\n<table id=\"Table_04_04_08\" class=\"unnumbered\" summary=\"The first column gives the following instructions of graphing a translation of f(x)=-log_b(x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the x-axis; 4. Draw a smooth curve through the points; 5. State the domain, (0, infinity), the range, (-infinity, infinity), and the vertical asymptote x=0. The second column gives the following instructions of graphing a translation of f(x)=log_b(-x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (-1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the y-axis; 4. Draw a smooth curve through the points; 5. State the domain, (-infinity, 0), the range, (-infinity, infinity), and the vertical asymptote x=0.\">\r\n<thead>\r\n<tr>\r\n<th>[latex]\\text{If }f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n<th>[latex]\\text{If }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1. Draw the vertical asymptote, <em>x\u00a0<\/em>= 0.<\/td>\r\n<td>1. Draw the vertical asymptote, <em>x\u00a0<\/em>= 0.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\r\n<td>2. Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis.<\/td>\r\n<td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y<\/em>-axis.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. Draw a smooth curve through the points.<\/td>\r\n<td>4. Draw a smooth curve through the points.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5. State the domain [latex]\\left(0,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\r\n<td>5. State the domain, [latex]\\left(-\\infty ,0\\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Reflection of a Logarithmic Function<\/h3>\r\nSketch 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.\r\n\r\n[reveal-answer q=\"843271\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843271\"]\r\n\r\nBefore graphing [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex], identify the behavior and key points for the graph.\r\n<ul>\r\n \t<li>Since <em>b\u00a0<\/em>= 10 is greater than one, we know that the parent function is increasing. Since the <em>input<\/em> value is multiplied by \u20131, <em>f<\/em>\u00a0is a reflection of the parent graph about the <em>y-<\/em>axis. Thus, [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] will be decreasing as <em>x<\/em>\u00a0moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>The <em>x<\/em>-intercept is [latex]\\left(-1,0\\right)[\/latex].<\/li>\r\n \t<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\r\n<\/ul>\r\n<figure id=\"CNX_Precalc_Figure_04_04_019\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233845\/CNX_Precalc_Figure_04_04_0192.jpg\" alt=\"Graph of two functions. The parent function is y=log(x), with an asymptote at x=0 and labeled points at (1, 0), and (10, 0).The translation function f(x)=log(-x) has an asymptote at x=0 and labeled points at (-1, 0) and (-10, 1).\" width=\"487\" height=\"363\" \/> The domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x\u00a0= 0.[\/caption]<\/figure>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGraph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right)[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"160849\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160849\"]\r\n\r\nThe domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.\r\n\r\n<img class=\"aligncenter size-full wp-image-3117\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16234244\/CNX_Precalc_Figure_04_04_0202.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" width=\"487\" height=\"288\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a logarithmic equation, use a graphing calculator to approximate solutions<\/h3>\r\n<ol>\r\n \t<li>Press <strong>[Y=]<\/strong>. Enter the given logarithmic equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\r\n \t<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\r\n \t<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \"intersect\" and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <i>x\u00a0<\/i>for the point(s) of intersection.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Approximating the Solution of a Logarithmic Equation<\/h3>\r\nSolve [latex]4\\mathrm{ln}\\left(x\\right)+1=-2\\mathrm{ln}\\left(x - 1\\right)[\/latex] graphically. Round to the nearest thousandth.\r\n\r\n[reveal-answer q=\"435068\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"435068\"]\r\n\r\nPress <strong>[Y=]<\/strong> and enter [latex]4\\mathrm{ln}\\left(x\\right)+1[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter [latex]-2\\mathrm{ln}\\left(x - 1\\right)[\/latex] next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for <em>x<\/em>\u00a0and \u201310 to 10 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to the right of <em>x\u00a0<\/em>= 1.\r\n\r\nFor 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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]5\\mathrm{log}\\left(x+2\\right)=4-\\mathrm{log}\\left(x\\right)[\/latex] graphically. Round to the nearest thousandth.\r\n\r\n[reveal-answer q=\"280798\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"280798\"]\r\n\r\n[latex]x\\approx 3.049[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summarizing Transformations of Logarithmic Functions<\/h2>\r\nNow that we have worked with each type of transformation for the logarithmic function, we can summarize each in the table below\u00a0to arrive at the general equation for transforming exponential functions.\r\n<table id=\"Table_04_04_009\" summary=\"Titled,\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\">Transformations 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>\r\n \t<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\r\n \t<li>Vertically <em>d<\/em>\u00a0units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Stretch and Compression\r\n<ul>\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>Reflection 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>Reflection 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 transformations<\/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 class=\"textbox\">\r\n<h3>A General Note: Transformations of Logarithmic Functions<\/h3>\r\nAll transformations of the parent logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] have the form\r\n\r\n[latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]\r\n\r\nwhere the parent function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right),b&gt;1[\/latex], is\r\n<ul>\r\n \t<li>shifted vertically up <em>d<\/em>\u00a0units.<\/li>\r\n \t<li>shifted horizontally to the left <em>c<\/em>\u00a0units.<\/li>\r\n \t<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\r\n \t<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\r\n \t<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\nFor [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.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Vertical Asymptote of A LogarithmIC Function<\/h3>\r\nWhat is the vertical asymptote of [latex]f\\left(x\\right)=-2{\\mathrm{log}}_{3}\\left(x+4\\right)+5[\/latex]?\r\n\r\n[reveal-answer q=\"714716\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"714716\"]\r\n\r\nThe vertical asymptote is at <em>x\u00a0<\/em>= \u20134.\r\n<h4>Analysis of the Solution<\/h4>\r\nThe coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to <em>x\u00a0<\/em>= \u20134.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWhat is the vertical asymptote of [latex]f\\left(x\\right)=3+\\mathrm{ln}\\left(x - 1\\right)[\/latex]?\r\n\r\n[reveal-answer q=\"502004\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"502004\"][latex]x=1[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nIn the example below, you'll write a common logarithmic function for the graph shown. Remember that all the functions studied in this course possess the characteristic that every point contained on the graph of a function satisfies the equation of the function.\r\n\r\nAs you have done before, begin with the form of a transformed logarithm function, [latex]f(x)=a\\text{log}(x+c)+d[\/latex], then fill in the parts you can discern from the graph.\r\n<ul>\r\n \t<li>Find the horizontal shift by locating the vertical asymptote.<\/li>\r\n \t<li>Examine the shape of the graph to see if it has been reflected.<\/li>\r\n \t<li>Once you have filled in what you know, substitute one or more points in integer coordinates if possible to solve for any remaining unknowns.<\/li>\r\n \t<li>Remember that if there are more than one unknown, you'll need more than one point and more than one equation to solve for all the unknowns.<\/li>\r\n<\/ul>\r\nWork through the example step-by-step with a pencil on paper, perhaps more than once or twice, to gain understanding.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Equation from a Graph<\/h3>\r\nFind a possible equation for the common logarithmic function graphed below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233847\/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\" \/>\r\n\r\n[reveal-answer q=\"671912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"671912\"]\r\n\r\nThis graph has a vertical asymptote at <em>x\u00a0<\/em>= \u20132 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have the form:\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+k[\/latex]<\/p>\r\nIt appears the graph passes through the points [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(2,-1\\right)[\/latex]. Substituting [latex]\\left(-1,1\\right)[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}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]<\/p>\r\nNext, substituting [latex]\\left(2,-1\\right)[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllll}-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]<\/p>\r\nThis gives us the equation [latex]f\\left(x\\right)=-\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can verify this answer by comparing the function values in the table below\u00a0with the points on the graph in this example.\r\n<table id=\"Table_04_04_010\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u22121<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<td>\u22120.58496<\/td>\r\n<td>\u22121<\/td>\r\n<td>\u22121.3219<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\r\n<td>\u22121.5850<\/td>\r\n<td>\u22121.8074<\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22122.1699<\/td>\r\n<td>\u22122.3219<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGive the equation of the natural logarithm graphed below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233849\/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\" \/>\r\n\r\n[reveal-answer q=\"752379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"752379\"]\r\n\r\n[latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x+3\\right)-1[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<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>\r\n\r\n<em>Yes if we know the function is a general logarithmic function. For example, look at the graph in the previous example. The graph approaches x = \u20133 (or thereabouts) more and more closely, so x = \u20133 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex]\\left\\{x|x&gt;-3\\right\\}[\/latex]. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is 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>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph stretches and compressions of logarithmic functions.<\/li>\n<li>Graph reflections of logarithmic functions.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Stretches and Compressions of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h2>\n<p>When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by a constant <em>a<\/em> &gt; 0, the result is a <strong>vertical stretch<\/strong> or <strong>compression<\/strong> of the original graph. To visualize stretches and compressions, we set <em>a\u00a0<\/em>&gt; 1 and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the vertical stretch, [latex]g\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], and the vertical compression, [latex]h\\left(x\\right)=\\frac{1}{a}{\\mathrm{log}}_{b}\\left(x\\right)[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Using an online graphing calculator plot the functions [latex]g(x) = a\\log_{b}{x}[\/latex]\u00a0 and\u00a0 [latex]h(x) = \\frac{1}{a}\\log_{b}{x}[\/latex]. One represents a vertical compression of the other. You may select any value for [latex]b[\/latex], though one between [latex]2[\/latex] and [latex]5[\/latex] will be easier to see. Experiment with various [latex]a[\/latex] values between [latex]1[\/latex] and [latex]10[\/latex]. As you investigate, consider the following questions:<\/p>\n<ul>\n<li>Both the vertical stretch and compression produce graphs that are increasing. Which transformation produces a function that increases faster?<\/li>\n<li>One of the key points that is commonly defined for transformations of a logarithmic function comes from finding the input that gives an output of [latex]y = 1[\/latex]. This point can help you determine whether a graph is the result of a vertical compression or stretch. Explain why.<\/li>\n<\/ul>\n<\/div>\n<p>The graphs below summarize the key features of the resulting graphs of vertical stretches and compressions of logarithmic functions.<\/p>\n<p><img decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233836\/CNX_Precalc_Figure_04_04_013n2.jpg\" alt=\"&quot;Graph\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Vertical Stretches and Compressions of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>For any constant <em>a<\/em> &gt; 1, the function [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/p>\n<ul>\n<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if <em>a\u00a0<\/em>&gt; 1.<\/li>\n<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if 0 &lt; <em>a\u00a0<\/em>&lt; 1.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>has the <em>x<\/em>-intercept [latex]\\left(1,0\\right)[\/latex].<\/li>\n<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a logarithmic function Of the form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], [latex]a>0[\/latex], graph the Stretch or Compression<\/h3>\n<ol>\n<li>Identify the vertical stretch or compression:\n<ul>\n<li>If [latex]|a|>1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is stretched by a factor of <em>a<\/em>\u00a0units.<\/li>\n<li>If [latex]|a|<1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is compressed by a factor of <em>a<\/em>\u00a0units.<\/li>\n<\/ul>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the <em>y<\/em>\u00a0coordinates in each point by <em>a<\/em>.<\/li>\n<li>Label the three points.<\/li>\n<li>The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x<\/em> = 0.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Logarithm functions follow the same principles as the other toolkit functions with regard to stretches, compressions, and reflections.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Stretch or Compression of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>Sketch the graph of [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q595868\">Show Solution<\/span><\/p>\n<div id=\"q595868\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the function is [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex], we will note that<i> <\/i>= 2.<\/p>\n<p>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>The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p>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>The new coordinates are found by multiplying the <em>y<\/em>\u00a0coordinates of each point by 2.<\/p>\n<p>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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233838\/CNX_Precalc_Figure_04_04_0142.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x\u00a0= 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q250125\">Show Solution<\/span><\/p>\n<div id=\"q250125\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3114\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16233042\/CNX_Precalc_Figure_04_04_0152.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1\/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).\" width=\"487\" height=\"364\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Combining a Shift and a Stretch<\/h3>\n<p>Sketch the graph of [latex]f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q804029\">Show Solution<\/span><\/p>\n<div id=\"q804029\" class=\"hidden-answer\" style=\"display: none\">\n<p>Remember, what happens inside parentheses happens first. First, we move the graph left 2 units and then stretch the function vertically by a factor of 5. The vertical asymptote will be shifted to <em>x\u00a0<\/em>= \u20132. The <em>x<\/em>-intercept will be [latex]\\left(-1,0\\right)[\/latex]. The domain will be [latex]\\left(-2,\\infty \\right)[\/latex]. Two points will help give the shape of the graph: [latex]\\left(-1,0\\right)[\/latex] and [latex]\\left(8,5\\right)[\/latex]. We chose <em>x\u00a0<\/em>= 8 as the <em>x<\/em>-coordinate of one point to graph because when <em>x\u00a0<\/em>= 8, <em>x\u00a0<\/em>+ 2 = 10, the base of the common logarithm.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233840\/CNX_Precalc_Figure_04_04_0162.jpg\" alt=\"Graph of three functions. The parent function is y=log(x), with an asymptote at x=0. The first translation function y=5log(x+2) has an asymptote at x=-2. The second translation function y=log(x+2) has an asymptote at x=-2.\" width=\"487\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(-2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x\u00a0= \u20132.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Sketch a graph of the function [latex]f\\left(x\\right)=3\\mathrm{log}\\left(x - 2\\right)+1[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q404704\">Show Solution<\/span><\/p>\n<div id=\"q404704\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3115\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16233717\/CNX_Precalc_Figure_04_04_0172.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" width=\"487\" height=\"439\" \/><\/p>\n<div id=\"fs-id1165137437228\" class=\"solution\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Graphing Reflections of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h2>\n<p>When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a <strong>reflection<\/strong> about the <em>x<\/em>-axis. When the <em>input<\/em> is multiplied by \u20131, the result is a reflection about the <em>y<\/em>-axis. To visualize reflections, we restrict <em>b\u00a0<\/em>&gt; 1 and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex], and the reflection about the <em>y<\/em>-axis, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Using an online graphing calculator, plot the functions [latex]f(x) = \\log_{b}{x},\\text{ }g(x)=-\\log_{b}{x},\\text{ and }h(x) = \\log_{b}({-x})[\/latex]. You may select any value for [latex]b[\/latex], though one between [latex]2[\/latex] and [latex]5[\/latex] will be easier to see. Also plot the point [latex](b,1)[\/latex]. Consider the following questions:<\/p>\n<ul>\n<li>Which graph, [latex]g(x) = -\\log_{b}{x} \\text{ or }h(x) = \\log_{b}({-x})[\/latex] represents a vertical reflection? \u00a0Which one represents a horizontal reflection?<\/li>\n<li>You already added the point [latex](b,1)[\/latex] as a point of interest for the function [latex]f(x)[\/latex]. Using the variable [latex]b[\/latex] as your [latex]x[\/latex] value, add the corresponding points of interest for [latex]g(x)\\text{ and }h(x)[\/latex].<\/li>\n<li>Does the vertical asymptote change when you reflect the graph of [latex]f(x)[\/latex] either vertically or horizontally?<\/li>\n<\/ul>\n<\/div>\n<p>The graphs below summarize the key characteristics of reflecting [latex]f(x) = \\log_{b}{x}[\/latex] horizontally and vertically.<\/p>\n<p><img decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233843\/CNX_Precalc_Figure_04_04_018n2.jpg\" alt=\"&quot;Graph\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Reflections of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>The function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex]<\/p>\n<ul>\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\u00a0<em>x\u00a0<\/em>= 0 which are unchanged from the parent function.<\/li>\n<\/ul>\n<p>The function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/p>\n<ul>\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\u00a0<em>x\u00a0<\/em>= 0 which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a logarithmic function with the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph a Reflection<\/h3>\n<table id=\"Table_04_04_08\" class=\"unnumbered\" summary=\"The first column gives the following instructions of graphing a translation of f(x)=-log_b(x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the x-axis; 4. Draw a smooth curve through the points; 5. State the domain, (0, infinity), the range, (-infinity, infinity), and the vertical asymptote x=0. The second column gives the following instructions of graphing a translation of f(x)=log_b(-x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (-1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the y-axis; 4. Draw a smooth curve through the points; 5. State the domain, (-infinity, 0), the range, (-infinity, infinity), and the vertical asymptote x=0.\">\n<thead>\n<tr>\n<th>[latex]\\text{If }f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\n<th>[latex]\\text{If }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1. Draw the vertical asymptote, <em>x\u00a0<\/em>= 0.<\/td>\n<td>1. Draw the vertical asymptote, <em>x\u00a0<\/em>= 0.<\/td>\n<\/tr>\n<tr>\n<td>2. Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\n<td>2. Plot the <em>x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>x<\/em>-axis.<\/td>\n<td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the <em>y<\/em>-axis.<\/td>\n<\/tr>\n<tr>\n<td>4. Draw a smooth curve through the points.<\/td>\n<td>4. Draw a smooth curve through the points.<\/td>\n<\/tr>\n<tr>\n<td>5. State the domain [latex]\\left(0,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\n<td>5. State the domain, [latex]\\left(-\\infty ,0\\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Reflection of a Logarithmic Function<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q843271\">Show Solution<\/span><\/p>\n<div id=\"q843271\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>\n<li>Since <em>b\u00a0<\/em>= 10 is greater than one, we know that the parent function is increasing. Since the <em>input<\/em> value is multiplied by \u20131, <em>f<\/em>\u00a0is a reflection of the parent graph about the <em>y-<\/em>axis. Thus, [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] will be decreasing as <em>x<\/em>\u00a0moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>The <em>x<\/em>-intercept is [latex]\\left(-1,0\\right)[\/latex].<\/li>\n<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\n<\/ul>\n<figure id=\"CNX_Precalc_Figure_04_04_019\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233845\/CNX_Precalc_Figure_04_04_0192.jpg\" alt=\"Graph of two functions. The parent function is y=log(x), with an asymptote at x=0 and labeled points at (1, 0), and (10, 0).The translation function f(x)=log(-x) has an asymptote at x=0 and labeled points at (-1, 0) and (-10, 1).\" width=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x\u00a0= 0.<\/p>\n<\/div>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q160849\">Show Solution<\/span><\/p>\n<div id=\"q160849\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3117\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16234244\/CNX_Precalc_Figure_04_04_0202.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" width=\"487\" height=\"288\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a logarithmic equation, use a graphing calculator to approximate solutions<\/h3>\n<ol>\n<li>Press <strong>[Y=]<\/strong>. Enter the given logarithmic equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\n<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\n<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select &#8220;intersect&#8221; and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <i>x\u00a0<\/i>for the point(s) of intersection.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Approximating the Solution of a Logarithmic Equation<\/h3>\n<p>Solve [latex]4\\mathrm{ln}\\left(x\\right)+1=-2\\mathrm{ln}\\left(x - 1\\right)[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q435068\">Show Solution<\/span><\/p>\n<div id=\"q435068\" class=\"hidden-answer\" style=\"display: none\">\n<p>Press <strong>[Y=]<\/strong> and enter [latex]4\\mathrm{ln}\\left(x\\right)+1[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter [latex]-2\\mathrm{ln}\\left(x - 1\\right)[\/latex] next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for <em>x<\/em>\u00a0and \u201310 to 10 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to the right of <em>x\u00a0<\/em>= 1.<\/p>\n<p>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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]5\\mathrm{log}\\left(x+2\\right)=4-\\mathrm{log}\\left(x\\right)[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q280798\">Show Solution<\/span><\/p>\n<div id=\"q280798\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x\\approx 3.049[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Summarizing Transformations of Logarithmic Functions<\/h2>\n<p>Now that we have worked with each type of transformation for the logarithmic function, we can summarize each in the table below\u00a0to arrive at the general equation for transforming exponential functions.<\/p>\n<table id=\"Table_04_04_009\" summary=\"Titled,\">\n<thead>\n<tr>\n<th colspan=\"2\">Transformations 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>\n<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\n<li>Vertically <em>d<\/em>\u00a0units up<\/li>\n<\/ul>\n<\/td>\n<td>[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Stretch and Compression<\/p>\n<ul>\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>Reflection 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>Reflection 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 transformations<\/td>\n<td>[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>A General Note: Transformations of Logarithmic Functions<\/h3>\n<p>All transformations of the parent logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] have the form<\/p>\n<p>[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/p>\n<p>where the parent function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right),b>1[\/latex], is<\/p>\n<ul>\n<li>shifted vertically up <em>d<\/em>\u00a0units.<\/li>\n<li>shifted horizontally to the left <em>c<\/em>\u00a0units.<\/li>\n<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\n<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\n<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<p>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 class=\"textbox exercises\">\n<h3>Example: Finding the Vertical Asymptote of A LogarithmIC Function<\/h3>\n<p>What is the vertical asymptote of [latex]f\\left(x\\right)=-2{\\mathrm{log}}_{3}\\left(x+4\\right)+5[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q714716\">Show Solution<\/span><\/p>\n<div id=\"q714716\" class=\"hidden-answer\" style=\"display: none\">\n<p>The vertical asymptote is at <em>x\u00a0<\/em>= \u20134.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to <em>x\u00a0<\/em>= \u20134.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>What is the vertical asymptote of [latex]f\\left(x\\right)=3+\\mathrm{ln}\\left(x - 1\\right)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q502004\">Show Solution<\/span><\/p>\n<div id=\"q502004\" class=\"hidden-answer\" style=\"display: none\">[latex]x=1[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>In the example below, you&#8217;ll write a common logarithmic function for the graph shown. Remember that all the functions studied in this course possess the characteristic that every point contained on the graph of a function satisfies the equation of the function.<\/p>\n<p>As you have done before, begin with the form of a transformed logarithm function, [latex]f(x)=a\\text{log}(x+c)+d[\/latex], then fill in the parts you can discern from the graph.<\/p>\n<ul>\n<li>Find the horizontal shift by locating the vertical asymptote.<\/li>\n<li>Examine the shape of the graph to see if it has been reflected.<\/li>\n<li>Once you have filled in what you know, substitute one or more points in integer coordinates if possible to solve for any remaining unknowns.<\/li>\n<li>Remember that if there are more than one unknown, you&#8217;ll need more than one point and more than one equation to solve for all the unknowns.<\/li>\n<\/ul>\n<p>Work through the example step-by-step with a pencil on paper, perhaps more than once or twice, to gain understanding.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Equation from a Graph<\/h3>\n<p>Find a possible equation for the common logarithmic function graphed below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233847\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q671912\">Show Solution<\/span><\/p>\n<div id=\"q671912\" class=\"hidden-answer\" style=\"display: none\">\n<p>This graph has a vertical asymptote at <em>x\u00a0<\/em>= \u20132 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have the form:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+k[\/latex]<\/p>\n<p>It appears the graph passes through the points [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(2,-1\\right)[\/latex]. Substituting [latex]\\left(-1,1\\right)[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}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]<\/p>\n<p>Next, substituting [latex]\\left(2,-1\\right)[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllll}-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]<\/p>\n<p>This gives us the equation [latex]f\\left(x\\right)=-\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can verify this answer by comparing the function values in the table below\u00a0with the points on the graph in this example.<\/p>\n<table id=\"Table_04_04_010\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>\u22120.58496<\/td>\n<td>\u22121<\/td>\n<td>\u22121.3219<\/td>\n<\/tr>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\n<td>\u22121.5850<\/td>\n<td>\u22121.8074<\/td>\n<td>\u22122<\/td>\n<td>\u22122.1699<\/td>\n<td>\u22122.3219<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Give the equation of the natural logarithm graphed below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233849\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q752379\">Show Solution<\/span><\/p>\n<div id=\"q752379\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x+3\\right)-1[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><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><em>Yes if we know the function is a general logarithmic function. For example, look at the graph in the previous example. The graph approaches x = \u20133 (or thereabouts) more and more closely, so x = \u20133 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex]\\left\\{x|x>-3\\right\\}[\/latex]. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is 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\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-273\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Reflections of Logarithmic Functions Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/5qmaorrlst\">https:\/\/www.desmos.com\/calculator\/5qmaorrlst<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Vertical Stretch and Compression of a Logarithmic Function Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/arlodqq7be\">https:\/\/www.desmos.com\/calculator\/arlodqq7be<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/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":395986,"menu_order":35,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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