{"id":1586,"date":"2015-11-12T18:35:28","date_gmt":"2015-11-12T18:35:28","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1586"},"modified":"2015-11-12T18:35:28","modified_gmt":"2015-11-12T18:35:28","slug":"graphing-transformations-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/graphing-transformations-of-logarithmic-functions\/","title":{"raw":"Graphing Transformations of Logarithmic Functions","rendered":"Graphing Transformations of Logarithmic Functions"},"content":{"raw":"<p id=\"fs-id1165137430986\">As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the <strong>parent function<\/strong> [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] without loss of shape.<\/p>\n\n<section id=\"fs-id1165137734884\" data-depth=\"2\"><h2 data-type=\"title\">Graphing a Horizontal Shift of\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/h2>\nWhen a constant <em>c<\/em>\u00a0is added to the input of the parent function [latex]f\\left(x\\right)=\\text{log}_{b}\\left(x\\right)\\\\[\/latex], the result is a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em data-effect=\"italics\">opposite<\/em> direction of the sign on <em>c<\/em>. To visualize horizontal shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] and for <em>c\u00a0<\/em>&gt; 0 alongside the shift left, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)\\\\[\/latex], and the shift right, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)\\\\[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201850\/CNX_Precalc_Figure_04_04_007n2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.\" width=\"900\" height=\"526\" data-media-type=\"image\/jpg\"\/><b>Figure 6<\/b>[\/caption]\n\n<div id=\"fs-id1165135296307\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Horizontal Shifts of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165135176174\">For any constant <em>c<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)\\\\[\/latex]<\/p>\n\n<ul id=\"fs-id1165135206192\"><li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\n\t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\n\t<li>has the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\n\t<li>has domain [latex]\\left(-c,\\infty \\right)\\\\[\/latex].<\/li>\n\t<li>has range [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex].<\/li>\n<\/ul><\/div>\n<div id=\"fs-id1165137641710\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165137641715\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)\\\\[\/latex], graph the translation.<\/h3>\n<ol id=\"fs-id1165137454284\" data-number-style=\"arabic\"><li>Identify the horizontal shift:\n<ol id=\"fs-id1165137454288\" data-number-style=\"lower-alpha\"><li>If <em>c<\/em> &gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] left <em>c<\/em>\u00a0units.<\/li>\n\t<li>If <em>c\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] right <em>c<\/em>\u00a0units.<\/li>\n<\/ol><\/li>\n\t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\n\t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting <em>c<\/em>\u00a0from the\u00a0<em>x<\/em>\u00a0coordinate.<\/li>\n\t<li>Label the three points.<\/li>\n\t<li>The Domain is [latex]\\left(-c,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u2013c.<\/li>\n<\/ol><\/div>\n<div id=\"Example_04_04_04\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137414959\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137414961\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 4:\u00a0Graphing a Horizontal Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137455420\">Sketch the horizontal shift [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137759883\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137759885\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)\\\\[\/latex], we notice [latex]x+\\left(-2\\right)=x - 2\\\\[\/latex].<\/p>\n<p id=\"fs-id1165137784630\">Thus <em>c\u00a0<\/em>= \u20132, so <em>c\u00a0<\/em>&lt; 0. This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\\\[\/latex] right 2 units.<\/p>\n<p id=\"fs-id1165137836995\">The vertical asymptote is [latex]x=-\\left(-2\\right)\\\\[\/latex] or <em>x\u00a0<\/em>= 2.<\/p>\n<p id=\"fs-id1165134042608\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)\\\\[\/latex], [latex]\\left(1,0\\right)\\\\[\/latex], and [latex]\\left(3,1\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165137475806\">The new coordinates are found by adding 2 to the <em>x<\/em>\u00a0coordinates.<\/p>\n<p id=\"fs-id1165137748449\">Label the points [latex]\\left(\\frac{7}{3},-1\\right)\\\\[\/latex], [latex]\\left(3,0\\right)\\\\[\/latex], and [latex]\\left(5,1\\right)\\\\[\/latex].<\/p>\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.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201852\/CNX_Precalc_Figure_04_04_0082.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).\" width=\"487\" height=\"363\" data-media-type=\"image\/jpg\"\/><b>Figure 7<\/b>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 4<\/h3>\n<p id=\"fs-id1165135329937\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<\/section><section id=\"fs-id1165135403538\" data-depth=\"2\"><h2 data-type=\"title\">Graphing a Vertical Shift of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h2>\nWhen a constant <em>d<\/em>\u00a0is added to the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex], the result is a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the direction of the sign on <em>d<\/em>. To visualize vertical shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] alongside the shift up, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\\\[\/latex] and the shift down, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d\\\\[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201853\/CNX_Precalc_Figure_04_04_010F2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.\" width=\"900\" height=\"684\" data-media-type=\"image\/jpg\"\/><b>Figure 8<\/b>[\/caption]\n\n<div id=\"fs-id1165137767601\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Vertical Shifts of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137661370\">For any constant <em>d<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\\\[\/latex]<\/p>\n\n<ul id=\"fs-id1165137803105\"><li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\n\t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\n\t<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n\t<li>has domain [latex]\\left(0,\\infty \\right)\\\\[\/latex].<\/li>\n\t<li>has range [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex].<\/li>\n<\/ul><\/div>\n<div id=\"fs-id1165137706002\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165137706009\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\\\[\/latex], graph the translation.<\/h3>\n<ol><li>Identify the vertical shift:\n<ol><li>If <em>d\u00a0<\/em>&gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] up <em>d<\/em>\u00a0units.<\/li>\n\t<li>If <em>d\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] down <em>d\u00a0<\/em>units.<\/li>\n<\/ol><\/li>\n\t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n\t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding <em>d<\/em>\u00a0to the <em>y\u00a0<\/em>coordinate.<\/li>\n\t<li>Label the three points.<\/li>\n\t<li>The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/li>\n<\/ol><\/div>\n<div id=\"Example_04_04_05\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137470057\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137470059\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 5: Graphing a Vertical Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137832038\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2\\\\[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137925369\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137465913\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2\\\\[\/latex], we will notice <em>d\u00a0<\/em>= \u20132. Thus <em>d\u00a0<\/em>&lt; 0.<\/p>\n<p id=\"fs-id1165135175015\">This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\\\[\/latex] down 2 units.<\/p>\n<p id=\"fs-id1165137644429\">The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p id=\"fs-id1165137408419\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)\\\\[\/latex], [latex]\\left(1,0\\right)\\\\[\/latex], and [latex]\\left(3,1\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165135503945\">The new coordinates are found by subtracting 2 from the <em data-effect=\"italics\">y <\/em>coordinates.<\/p>\n<p id=\"fs-id1165135421660\">Label the points [latex]\\left(\\frac{1}{3},-3\\right)\\\\[\/latex], [latex]\\left(1,-2\\right)\\\\[\/latex], and [latex]\\left(3,-1\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165135195524\">The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x<\/em> = 0.<span id=\"fs-id1165134393856\" data-type=\"media\" data-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).\">\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201855\/CNX_Precalc_Figure_04_04_0112.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 1).\" data-media-type=\"image\/jpg\"\/><\/span><\/p>\n<p id=\"fs-id1165137698285\" style=\"text-align: center;\"><strong>Figure 9.\u00a0<\/strong>The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x<\/em> = 0.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 5<\/h3>\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<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<\/section><section id=\"fs-id1165137770245\" data-depth=\"2\"><h2 data-type=\"title\">Graphing Stretches and Compressions of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h2>\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].\n\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201856\/CNX_Precalc_Figure_04_04_013n2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=alog_b(x) when a&gt;1 is the translation function with an asymptote at x=0. The graph note the intersection of the two lines at (1, 0). This shows the translation of a vertical stretch.\" width=\"900\" height=\"700\" data-media-type=\"image\/jpg\"\/><b>Figure 10<\/b>[\/caption]\n\n<div id=\"fs-id1165137433996\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Vertical Stretches and Compressions of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137758179\">For any constant <em>a<\/em> &gt; 1, the function [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/p>\n\n<ul id=\"fs-id1165137428102\"><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\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>\n\t<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n\t<li>has the <em data-effect=\"italics\">x<\/em>-intercept [latex]\\left(1,0\\right)\\\\[\/latex].<\/li>\n\t<li>has domain [latex]\\left(0,\\infty \\right)\\\\[\/latex].<\/li>\n\t<li>has range [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex].<\/li>\n<\/ul><\/div>\n<div id=\"fs-id1165135169301\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165135169307\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex], [latex]a&gt;0\\\\[\/latex], graph the translation.<\/h3>\n<ol id=\"fs-id1165137464127\" data-number-style=\"arabic\"><li>Identify the vertical stretch or compressions:\n<ol id=\"eip-id1165134081434\"><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>\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>\n<\/ol><\/li>\n\t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n\t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the <em>y<\/em>\u00a0coordinates by <em>a<\/em>.<\/li>\n\t<li>Label the three points.<\/li>\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>\n<\/ol><\/div>\n<div id=\"Example_04_04_06\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135309914\" class=\"exercise\" data-type=\"exercise\">\n<div class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 6: Graphing a Stretch or Compression of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137602128\">Sketch a graph of [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135210050\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165135210052\">Since the function is [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)\\\\[\/latex], we will notice <em>a\u00a0<\/em>= 2.<\/p>\n<p id=\"fs-id1165135384321\">This means we will stretch the function [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\\\[\/latex] by a factor of 2.<\/p>\n<p id=\"fs-id1165135481989\">The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p id=\"fs-id1165137757801\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{4},-1\\right)\\\\[\/latex], [latex]\\left(1,0\\right)\\\\[\/latex], and [latex]\\left(4,1\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165135570058\">The new coordinates are found by multiplying the <em>y<\/em>\u00a0coordinates by 2.<\/p>\n<p id=\"fs-id1165137837989\">Label the points [latex]\\left(\\frac{1}{4},-2\\right)\\\\[\/latex], [latex]\\left(1,0\\right)\\\\[\/latex], and [latex]\\left(4,\\text{2}\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165135543469\">The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<span id=\"fs-id1165134059742\" data-type=\"media\" data-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).\">\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201857\/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).\" data-media-type=\"image\/jpg\"\/><\/span><\/p>\n<p id=\"fs-id1165135566827\" style=\"text-align: center;\"><strong>Figure 11.\u00a0<\/strong>The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 6<\/h3>\n<p id=\"fs-id1165135471122\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\mathrm{log}}_{4}\\left(x\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<div id=\"Example_04_04_07\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165134267814\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134267816\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 7: Combining a Shift and a Stretch<\/h3>\n<p id=\"fs-id1165137863045\">Sketch a graph of [latex]f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\right)\\\\[\/latex]. State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137935559\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137935561\">Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5. The vertical asymptote will be shifted to <em>x\u00a0<\/em>= \u20132. The <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em>-coordinate of one point to graph because when <em>x\u00a0<\/em>= 8, <em>x\u00a0<\/em>+ 2 = 10, the base of the common logarithm.<span id=\"fs-id1165135641650\" data-type=\"media\" data-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.\">\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201859\/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.\" data-media-type=\"image\/jpg\"\/><\/span><\/p>\n<p id=\"fs-id1165137874883\" style=\"text-align: center;\"><strong>Figure 12.\u00a0<\/strong>The domain is [latex]\\left(-2,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u20132.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 7<\/h3>\n<p id=\"fs-id1165137838697\">Sketch a graph of the function [latex]f\\left(x\\right)=3\\mathrm{log}\\left(x - 2\\right)+1\\\\[\/latex]. State the domain, range, and asymptote.<\/p>\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<\/section><section id=\"fs-id1165137629003\" data-depth=\"2\"><h2>Graphing Reflections of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/h2>\n<p id=\"fs-id1165135169315\">When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] is multiplied by \u20131, the result is a <strong>reflection<\/strong> about the <em data-effect=\"italics\">x<\/em>-axis. When the <em data-effect=\"italics\">input<\/em> is multiplied by \u20131, the result is a reflection about the <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em>-axis, [latex]g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)\\\\[\/latex] and the reflection about the <em data-effect=\"italics\">y<\/em>-axis, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)\\\\[\/latex].<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"901\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201900\/CNX_Precalc_Figure_04_04_018n2.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) when b&gt;1 is the translation function with an asymptote at x=0. The graph note the intersection of the two lines at (1, 0). This shows the translation of a reflection about the x-axis.\" width=\"901\" height=\"726\" data-media-type=\"image\/jpg\"\/><b>Figure 13<\/b>[\/caption]\n\n<div id=\"fs-id1165135190744\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Reflections of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137722409\">The function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)\\\\[\/latex]<\/p>\n\n<ul id=\"fs-id1165137832285\"><li>reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\n\t<li>has domain, [latex]\\left(0,\\infty \\right)\\\\[\/latex], range, [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and vertical asymptote, <em>x\u00a0<\/em>= 0, which are unchanged from the parent function.<\/li>\n<\/ul>\nThe function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)\\\\[\/latex]\n<ul id=\"fs-id1165137734930\"><li>reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\n\t<li>has domain [latex]\\left(-\\infty ,0\\right)\\\\[\/latex].<\/li>\n\t<li>has range, [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and vertical asymptote, <em>x\u00a0<\/em>= 0, which are unchanged from the parent function.<\/li>\n<\/ul><\/div>\n<div id=\"fs-id1165137638830\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165137638837\">How To: Given a logarithmic function with the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex], graph a translation.<\/h3>\n<table id=\"Table_04_04_08\" class=\"unnumbered\" summary=\"The first column gives the following instructions of graphing a translation of f(x)=-log_b(x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the x-axis; 4. Draw a smooth curve through the points; 5. State the domain, (0, infinity), the range, (-infinity, infinity), and the vertical asymptote x=0. The second column gives the following instructions of graphing a translation of f(x)=log_b(-x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (-1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the y-axis; 4. Draw a smooth curve through the points; 5. State the domain, (-infinity, 0), the range, (-infinity, infinity), and the vertical asymptote x=0.\" data-frame=\"none\" data-label=\"\"><thead><tr><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><\/thead><tbody><tr><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><tr><td>2. Plot the <em data-effect=\"italics\">x-<\/em>intercept, [latex]\\left(1,0\\right)\\\\[\/latex].<\/td>\n<td>2. Plot the <em data-effect=\"italics\">x-<\/em>intercept, [latex]\\left(1,0\\right)\\\\[\/latex].<\/td>\n<\/tr><tr><td>3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] about the <em data-effect=\"italics\">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 data-effect=\"italics\">y<\/em>-axis.<\/td>\n<\/tr><tr><td>4. Draw a smooth curve through the points.<\/td>\n<td>4. Draw a smooth curve through the points.<\/td>\n<\/tr><tr><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><\/tbody><\/table><\/div>\n<div id=\"Example_04_04_08\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137697928\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137849033\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 8: Graphing a Reflection of a Logarithmic Function<\/h3>\n<p id=\"fs-id1165137849038\">Sketch a graph of [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137836523\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\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\n<ul id=\"fs-id1165137769879\"><li>Since <em>b\u00a0<\/em>= 10 is greater than one, we know that the parent function is increasing. Since the <em data-effect=\"italics\">input<\/em> value is multiplied by \u20131, <em>f<\/em>\u00a0is a reflection of the parent graph about the <em data-effect=\"italics\">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\t<li>The <em data-effect=\"italics\">x<\/em>-intercept is [latex]\\left(-1,0\\right)\\\\[\/latex].<\/li>\n\t<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\n<\/ul><figure id=\"CNX_Precalc_Figure_04_04_019\" class=\"small\"><span id=\"fs-id1165134042188\" data-type=\"media\" data-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).\"> <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201901\/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).\" data-media-type=\"image\/jpg\"\/><\/span><\/figure><p id=\"fs-id1165134042202\" style=\"text-align: center;\"><strong>Figure 14.\u00a0<\/strong>The domain is [latex]\\left(-\\infty ,0\\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 8<\/h3>\n<p id=\"fs-id1165135681852\">Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right)\\\\[\/latex]. State the domain, range, and asymptote.<\/p>\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<div id=\"fs-id1165134579621\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165134579627\">How To: Given a logarithmic equation, use a graphing calculator to approximate solutions.<\/h3>\n<ol id=\"fs-id1165137431118\" data-number-style=\"arabic\"><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\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>\n\t<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \"intersect\" and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x<\/em>, for the point(s) of intersection.<\/li>\n<\/ol><\/div>\n<div id=\"Example_04_04_09\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135414229\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135414231\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 9: Approximating the Solution of a Logarithmic Equation<\/h3>\n<p id=\"fs-id1165135414236\">Solve [latex]4\\mathrm{ln}\\left(x\\right)+1=-2\\mathrm{ln}\\left(x - 1\\right)\\\\[\/latex] graphically. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135193432\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165135193434\">Press <strong>[Y=]<\/strong> and enter [latex]4\\mathrm{ln}\\left(x\\right)+1\\\\[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter [latex]-2\\mathrm{ln}\\left(x - 1\\right)\\\\[\/latex] next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for <em>x<\/em>\u00a0and \u201310 to 10 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to right of <em>x\u00a0<\/em>= 1.<\/p>\n<p id=\"fs-id1165135245763\">For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em data-effect=\"italics\">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\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 9<\/h3>\n<p id=\"fs-id1165137639531\">Solve [latex]5\\mathrm{log}\\left(x+2\\right)=4-\\mathrm{log}\\left(x\\right)\\\\[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<\/section><section id=\"fs-id1165135528930\" data-depth=\"2\"><h2 data-type=\"title\">Summarizing Translations of the Logarithmic Function<\/h2>\n<p id=\"fs-id1165135528935\">Now that we have worked with each type of translation for the logarithmic function, we can summarize each in the table below\u00a0to arrive at the general equation for translating exponential functions.<\/p>\n\n<table id=\"Table_04_04_009\" summary=\"Titled,\"><thead><tr><th style=\"text-align: center;\" colspan=\"2\">Translations of the Parent Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/th>\n<\/tr><tr><th style=\"text-align: center;\">Translation<\/th>\n<th style=\"text-align: center;\">Form<\/th>\n<\/tr><\/thead><tbody><tr><td>Shift\n<ul id=\"fs-id1165137416971\"><li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\n\t<li>Vertically <em>d<\/em>\u00a0units up<\/li>\n<\/ul><\/td>\n<td data-align=\"left\">[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d\\\\[\/latex]<\/td>\n<\/tr><tr><td>Stretch and Compress\n<ul id=\"fs-id1165137427553\"><li>Stretch if [latex]|a|&gt;1\\\\[\/latex]<\/li>\n\t<li>Compression if [latex]|a|&lt;1\\\\[\/latex]<\/li>\n<\/ul><\/td>\n<td data-align=\"left\">[latex]y=a{\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/td>\n<\/tr><tr><td>Reflect about the <em data-effect=\"italics\">x<\/em>-axis<\/td>\n<td data-align=\"left\">[latex]y=-{\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/td>\n<\/tr><tr><td>Reflect about the <em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td data-align=\"left\">[latex]y={\\mathrm{log}}_{b}\\left(-x\\right)\\\\[\/latex]<\/td>\n<\/tr><tr><td>General equation for all translations<\/td>\n<td data-align=\"left\">[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d\\\\[\/latex]<\/td>\n<\/tr><\/tbody><\/table><div id=\"fs-id1165137414493\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Translations of Logarithmic Functions<\/h3>\n<p id=\"fs-id1165137414501\">All translations of the parent logarithmic function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex], have the form<\/p>\n\n<div id=\"fs-id1165135408512\" class=\"equation\" style=\"text-align: center;\" data-type=\"equation\">[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&gt;1\\\\[\/latex], is<\/p>\n\n<ul id=\"fs-id1165137531610\"><li>shifted vertically up <em>d<\/em>\u00a0units.<\/li>\n\t<li>shifted horizontally to the left <em>c<\/em>\u00a0units.<\/li>\n\t<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\n\t<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\n\t<li>reflected about the <em data-effect=\"italics\">x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\n<\/ul><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 data-effect=\"italics\">y<\/em>-axis.<\/p>\n\n<\/div>\n<div id=\"Example_04_04_10\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135296269\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135296271\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 10: Finding the Vertical Asymptote of a Logarithm Graph<\/h3>\n<p id=\"fs-id1165135296276\">What is the vertical asymptote of [latex]f\\left(x\\right)=-2{\\mathrm{log}}_{3}\\left(x+4\\right)+5\\\\[\/latex]?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137572550\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137572552\">The vertical asymptote is at <em>x\u00a0<\/em>= \u20134.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137871955\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165137871960\">The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to <em>x\u00a0<\/em>= \u20134.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 10<\/h3>\n<p id=\"fs-id1165135368433\">What is the vertical asymptote of [latex]f\\left(x\\right)=3+\\mathrm{ln}\\left(x - 1\\right)\\\\[\/latex]?<\/p>\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<div id=\"Example_04_04_11\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137849555\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137849558\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 11: Finding the Equation from a Graph<\/h3>\n<p id=\"fs-id1165137849563\">Find a possible equation for the common logarithmic function graphed in Figure 15.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201903\/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\" data-media-type=\"image\/jpg\"\/><b>Figure 15<\/b>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135342977\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165135342979\">This graph has a vertical asymptote at <em>x\u00a0<\/em>= \u20132 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:<\/p>\n\n<div id=\"eip-id1165133361454\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[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(-1,1\\right)\\\\[\/latex] and [latex]\\left(2,-1\\right)\\\\[\/latex]. Substituting [latex]\\left(-1,1\\right)\\\\[\/latex],<\/p>\n\n<div id=\"eip-id1165134101923\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}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{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137628655\">Next, substituting in [latex]\\left(2,-1\\right)\\\\[\/latex],<\/p>\n\n<div id=\"eip-id1165135431720\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}-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{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135192211\">This gives us the equation [latex]f\\left(x\\right)=-\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1\\\\[\/latex].<\/p>\n\n<\/div>\n<div id=\"fs-id1165137735581\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165137735586\">We can verify this answer by comparing the function values in the table below\u00a0with the points on the graph in Example 11.<\/p>\n\n<table id=\"Table_04_04_010\" summary=\"..\"><colgroup><col data-align=\"center\"\/><col data-align=\"center\"\/><\/colgroup><tbody><tr><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><tr><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><tr><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><tr><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><\/tbody><\/table><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 11<\/h3>\n<p id=\"fs-id1165137665487\">Give the equation of the natural logarithm graphed in Figure 16.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201904\/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\" data-media-type=\"image\/jpg\"\/><b>Figure 16<\/b>[\/caption]\n\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>\n<div id=\"fs-id1165137855236\" class=\"note precalculus qa textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"Q&amp;A\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165137855242\"><strong>Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?<\/strong><\/p>\n<p id=\"fs-id1165137827126\"><em data-effect=\"italics\">Yes, if we know the function is a general logarithmic function. For example, look at the graph in Try It 11. The graph approaches x = \u20133 (or thereabouts) more and more closely, so x = \u20133 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex]\\left\\{x|x&gt;-3\\right\\}\\\\[\/latex]. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as [latex]x\\to -{3}^{+},f\\left(x\\right)\\to -\\infty \\\\[\/latex] and as [latex]x\\to \\infty ,f\\left(x\\right)\\to \\infty \\\\[\/latex].<\/em><\/p>\n\n<\/div>\n<\/section>","rendered":"<p id=\"fs-id1165137430986\">As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the <strong>parent function<\/strong> [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] without loss of shape.<\/p>\n<section id=\"fs-id1165137734884\" data-depth=\"2\">\n<h2 data-type=\"title\">Graphing a Horizontal Shift of\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/h2>\n<p>When a constant <em>c<\/em>\u00a0is added to the input of the parent function [latex]f\\left(x\\right)=\\text{log}_{b}\\left(x\\right)\\\\[\/latex], the result is a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em data-effect=\"italics\">opposite<\/em> direction of the sign on <em>c<\/em>. To visualize horizontal shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] and for <em>c\u00a0<\/em>&gt; 0 alongside the shift left, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)\\\\[\/latex], and the shift right, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)\\\\[\/latex].<\/p>\n<div style=\"width: 910px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201850\/CNX_Precalc_Figure_04_04_007n2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.\" width=\"900\" height=\"526\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165135296307\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Horizontal Shifts of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165135176174\">For any constant <em>c<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)\\\\[\/latex]<\/p>\n<ul id=\"fs-id1165135206192\">\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\n<li>has domain [latex]\\left(-c,\\infty \\right)\\\\[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137641710\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165137641715\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)\\\\[\/latex], graph the translation.<\/h3>\n<ol id=\"fs-id1165137454284\" data-number-style=\"arabic\">\n<li>Identify the horizontal shift:\n<ol id=\"fs-id1165137454288\" data-number-style=\"lower-alpha\">\n<li>If <em>c<\/em> &gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] left <em>c<\/em>\u00a0units.<\/li>\n<li>If <em>c\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] right <em>c<\/em>\u00a0units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting <em>c<\/em>\u00a0from the\u00a0<em>x<\/em>\u00a0coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The Domain is [latex]\\left(-c,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u2013c.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_04\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137414959\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137414961\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 4:\u00a0Graphing a Horizontal Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137455420\">Sketch the horizontal shift [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137759883\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137759885\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)\\\\[\/latex], we notice [latex]x+\\left(-2\\right)=x - 2\\\\[\/latex].<\/p>\n<p id=\"fs-id1165137784630\">Thus <em>c\u00a0<\/em>= \u20132, so <em>c\u00a0<\/em>&lt; 0. This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\\\[\/latex] right 2 units.<\/p>\n<p id=\"fs-id1165137836995\">The vertical asymptote is [latex]x=-\\left(-2\\right)\\\\[\/latex] or <em>x\u00a0<\/em>= 2.<\/p>\n<p id=\"fs-id1165134042608\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)\\\\[\/latex], [latex]\\left(1,0\\right)\\\\[\/latex], and [latex]\\left(3,1\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165137475806\">The new coordinates are found by adding 2 to the <em>x<\/em>\u00a0coordinates.<\/p>\n<p id=\"fs-id1165137748449\">Label the points [latex]\\left(\\frac{7}{3},-1\\right)\\\\[\/latex], [latex]\\left(3,0\\right)\\\\[\/latex], and [latex]\\left(5,1\\right)\\\\[\/latex].<\/p>\n<p>The domain is [latex]\\left(2,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201852\/CNX_Precalc_Figure_04_04_0082.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).\" width=\"487\" height=\"363\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 4<\/h3>\n<p id=\"fs-id1165135329937\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135403538\" data-depth=\"2\">\n<h2 data-type=\"title\">Graphing a Vertical Shift of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h2>\n<p>When a constant <em>d<\/em>\u00a0is added to the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex], the result is a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the direction of the sign on <em>d<\/em>. To visualize vertical shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] alongside the shift up, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\\\[\/latex] and the shift down, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d\\\\[\/latex].<\/p>\n<div style=\"width: 910px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201853\/CNX_Precalc_Figure_04_04_010F2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.\" width=\"900\" height=\"684\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165137767601\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Vertical Shifts of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137661370\">For any constant <em>d<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\\\[\/latex]<\/p>\n<ul id=\"fs-id1165137803105\">\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>has domain [latex]\\left(0,\\infty \\right)\\\\[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137706002\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165137706009\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d\\\\[\/latex], graph the translation.<\/h3>\n<ol>\n<li>Identify the vertical shift:\n<ol>\n<li>If <em>d\u00a0<\/em>&gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] up <em>d<\/em>\u00a0units.<\/li>\n<li>If <em>d\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] down <em>d\u00a0<\/em>units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding <em>d<\/em>\u00a0to the <em>y\u00a0<\/em>coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_05\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137470057\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137470059\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 5: Graphing a Vertical Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137832038\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2\\\\[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137925369\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137465913\">Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2\\\\[\/latex], we will notice <em>d\u00a0<\/em>= \u20132. Thus <em>d\u00a0<\/em>&lt; 0.<\/p>\n<p id=\"fs-id1165135175015\">This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\\\[\/latex] down 2 units.<\/p>\n<p id=\"fs-id1165137644429\">The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p id=\"fs-id1165137408419\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)\\\\[\/latex], [latex]\\left(1,0\\right)\\\\[\/latex], and [latex]\\left(3,1\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165135503945\">The new coordinates are found by subtracting 2 from the <em data-effect=\"italics\">y <\/em>coordinates.<\/p>\n<p id=\"fs-id1165135421660\">Label the points [latex]\\left(\\frac{1}{3},-3\\right)\\\\[\/latex], [latex]\\left(1,-2\\right)\\\\[\/latex], and [latex]\\left(3,-1\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165135195524\">The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x<\/em> = 0.<span id=\"fs-id1165134393856\" data-type=\"media\" data-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).\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201855\/CNX_Precalc_Figure_04_04_0112.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 1).\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165137698285\" style=\"text-align: center;\"><strong>Figure 9.\u00a0<\/strong>The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x<\/em> = 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 5<\/h3>\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<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137770245\" data-depth=\"2\">\n<h2 data-type=\"title\">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 style=\"width: 910px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201856\/CNX_Precalc_Figure_04_04_013n2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=alog_b(x) when a&gt;1 is the translation function with an asymptote at x=0. The graph note the intersection of the two lines at (1, 0). This shows the translation of a vertical stretch.\" width=\"900\" height=\"700\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165137433996\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Vertical Stretches and Compressions of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137758179\">For any constant <em>a<\/em> &gt; 1, the function [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/p>\n<ul id=\"fs-id1165137428102\">\n<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] vertically by a factor of <em>a<\/em>\u00a0if <em>a\u00a0<\/em>&gt; 1.<\/li>\n<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] vertically by a factor of <em>a<\/em>\u00a0if 0 &lt; <em>a\u00a0<\/em>&lt; 1.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>has the <em data-effect=\"italics\">x<\/em>-intercept [latex]\\left(1,0\\right)\\\\[\/latex].<\/li>\n<li>has domain [latex]\\left(0,\\infty \\right)\\\\[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135169301\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165135169307\">How To: Given a logarithmic function with the form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex], [latex]a>0\\\\[\/latex], graph the translation.<\/h3>\n<ol id=\"fs-id1165137464127\" data-number-style=\"arabic\">\n<li>Identify the vertical stretch or compressions:\n<ol id=\"eip-id1165134081434\">\n<li>If [latex]|a|>1\\\\[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] is stretched by a factor of <em>a<\/em>\u00a0units.<\/li>\n<li>If [latex]|a|<1\\\\[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] is compressed by a factor of <em>a<\/em>\u00a0units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the <em>y<\/em>\u00a0coordinates by <em>a<\/em>.<\/li>\n<li>Label the three points.<\/li>\n<li>The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x<\/em> = 0.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_06\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135309914\" class=\"exercise\" data-type=\"exercise\">\n<div class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 6: Graphing a Stretch or Compression of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137602128\">Sketch a graph of [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165135210050\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165135210052\">Since the function is [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)\\\\[\/latex], we will notice <em>a\u00a0<\/em>= 2.<\/p>\n<p id=\"fs-id1165135384321\">This means we will stretch the function [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\\\[\/latex] by a factor of 2.<\/p>\n<p id=\"fs-id1165135481989\">The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p id=\"fs-id1165137757801\">Consider the three key points from the parent function, [latex]\\left(\\frac{1}{4},-1\\right)\\\\[\/latex], [latex]\\left(1,0\\right)\\\\[\/latex], and [latex]\\left(4,1\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165135570058\">The new coordinates are found by multiplying the <em>y<\/em>\u00a0coordinates by 2.<\/p>\n<p id=\"fs-id1165137837989\">Label the points [latex]\\left(\\frac{1}{4},-2\\right)\\\\[\/latex], [latex]\\left(1,0\\right)\\\\[\/latex], and [latex]\\left(4,\\text{2}\\right)\\\\[\/latex].<\/p>\n<p id=\"fs-id1165135543469\">The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<span id=\"fs-id1165134059742\" data-type=\"media\" data-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).\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201857\/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).\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165135566827\" style=\"text-align: center;\"><strong>Figure 11.\u00a0<\/strong>The domain is [latex]\\left(0,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 6<\/h3>\n<p id=\"fs-id1165135471122\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\mathrm{log}}_{4}\\left(x\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<div id=\"Example_04_04_07\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165134267814\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134267816\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 7: Combining a Shift and a Stretch<\/h3>\n<p id=\"fs-id1165137863045\">Sketch a graph of [latex]f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\right)\\\\[\/latex]. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137935559\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137935561\">Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5. The vertical asymptote will be shifted to <em>x\u00a0<\/em>= \u20132. The <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em>-coordinate of one point to graph because when <em>x\u00a0<\/em>= 8, <em>x\u00a0<\/em>+ 2 = 10, the base of the common logarithm.<span id=\"fs-id1165135641650\" data-type=\"media\" data-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.\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201859\/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.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165137874883\" style=\"text-align: center;\"><strong>Figure 12.\u00a0<\/strong>The domain is [latex]\\left(-2,\\infty \\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u20132.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 7<\/h3>\n<p id=\"fs-id1165137838697\">Sketch a graph of the function [latex]f\\left(x\\right)=3\\mathrm{log}\\left(x - 2\\right)+1\\\\[\/latex]. State the domain, range, and asymptote.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137629003\" data-depth=\"2\">\n<h2>Graphing Reflections of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/h2>\n<p id=\"fs-id1165135169315\">When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] is multiplied by \u20131, the result is a <strong>reflection<\/strong> about the <em data-effect=\"italics\">x<\/em>-axis. When the <em data-effect=\"italics\">input<\/em> is multiplied by \u20131, the result is a reflection about the <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em>-axis, [latex]g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)\\\\[\/latex] and the reflection about the <em data-effect=\"italics\">y<\/em>-axis, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)\\\\[\/latex].<\/p>\n<div style=\"width: 911px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201900\/CNX_Precalc_Figure_04_04_018n2.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) when b&gt;1 is the translation function with an asymptote at x=0. The graph note the intersection of the two lines at (1, 0). This shows the translation of a reflection about the x-axis.\" width=\"901\" height=\"726\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165135190744\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Reflections of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)\\\\[\/latex]<\/h3>\n<p id=\"fs-id1165137722409\">The function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)\\\\[\/latex]<\/p>\n<ul id=\"fs-id1165137832285\">\n<li>reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\n<li>has domain, [latex]\\left(0,\\infty \\right)\\\\[\/latex], range, [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and vertical asymptote, <em>x\u00a0<\/em>= 0, which are unchanged from the parent function.<\/li>\n<\/ul>\n<p>The function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)\\\\[\/latex]<\/p>\n<ul id=\"fs-id1165137734930\">\n<li>reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\n<li>has domain [latex]\\left(-\\infty ,0\\right)\\\\[\/latex].<\/li>\n<li>has range, [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and vertical asymptote, <em>x\u00a0<\/em>= 0, which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137638830\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165137638837\">How To: Given a logarithmic function with the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex], graph a translation.<\/h3>\n<table id=\"Table_04_04_08\" class=\"unnumbered\" summary=\"The first column gives the following instructions of graphing a translation of f(x)=-log_b(x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the x-axis; 4. Draw a smooth curve through the points; 5. State the domain, (0, infinity), the range, (-infinity, infinity), and the vertical asymptote x=0. The second column gives the following instructions of graphing a translation of f(x)=log_b(-x) with the parent function being f(x)=log_b(x): 1. Draw the vertical asymptote, x=0; 2. Plot the x-intercept, (-1, 0); 3. Reflect the graph of the parent function f(x)=log_b(x) about the y-axis; 4. Draw a smooth curve through the points; 5. State the domain, (-infinity, 0), the range, (-infinity, infinity), and the vertical asymptote x=0.\" data-frame=\"none\" data-label=\"\">\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 data-effect=\"italics\">x-<\/em>intercept, [latex]\\left(1,0\\right)\\\\[\/latex].<\/td>\n<td>2. Plot the <em data-effect=\"italics\">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 data-effect=\"italics\">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 data-effect=\"italics\">y<\/em>-axis.<\/td>\n<\/tr>\n<tr>\n<td>4. Draw a smooth curve through the points.<\/td>\n<td>4. Draw a smooth curve through the points.<\/td>\n<\/tr>\n<tr>\n<td>5. State the domain, [latex]\\left(0,\\infty \\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\n<td>5. State the domain, [latex]\\left(-\\infty ,0\\right)\\\\[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote <em>x\u00a0<\/em>= 0.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"Example_04_04_08\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137697928\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137849033\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 8: Graphing a Reflection of a Logarithmic Function<\/h3>\n<p id=\"fs-id1165137849038\">Sketch a graph of [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)\\\\[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137836523\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137836525\">Before graphing [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)\\\\[\/latex], identify the behavior and key points for the graph.<\/p>\n<ul id=\"fs-id1165137769879\">\n<li>Since <em>b\u00a0<\/em>= 10 is greater than one, we know that the parent function is increasing. Since the <em data-effect=\"italics\">input<\/em> value is multiplied by \u20131, <em>f<\/em>\u00a0is a reflection of the parent graph about the <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em>-intercept is [latex]\\left(-1,0\\right)\\\\[\/latex].<\/li>\n<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\n<\/ul>\n<figure id=\"CNX_Precalc_Figure_04_04_019\" class=\"small\"><span id=\"fs-id1165134042188\" data-type=\"media\" data-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).\"> <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201901\/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).\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<p id=\"fs-id1165134042202\" style=\"text-align: center;\"><strong>Figure 14.\u00a0<\/strong>The domain is [latex]\\left(-\\infty ,0\\right)\\\\[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)\\\\[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 8<\/h3>\n<p id=\"fs-id1165135681852\">Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right)\\\\[\/latex]. State the domain, range, and asymptote.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<div id=\"fs-id1165134579621\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165134579627\">How To: Given a logarithmic equation, use a graphing calculator to approximate solutions.<\/h3>\n<ol id=\"fs-id1165137431118\" data-number-style=\"arabic\">\n<li>Press <strong>[Y=]<\/strong>. Enter the given logarithm equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\n<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\n<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select &#8220;intersect&#8221; and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x<\/em>, for the point(s) of intersection.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_04_09\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135414229\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135414231\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 9: Approximating the Solution of a Logarithmic Equation<\/h3>\n<p id=\"fs-id1165135414236\">Solve [latex]4\\mathrm{ln}\\left(x\\right)+1=-2\\mathrm{ln}\\left(x - 1\\right)\\\\[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165135193432\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165135193434\">Press <strong>[Y=]<\/strong> and enter [latex]4\\mathrm{ln}\\left(x\\right)+1\\\\[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter [latex]-2\\mathrm{ln}\\left(x - 1\\right)\\\\[\/latex] next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for <em>x<\/em>\u00a0and \u201310 to 10 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to right of <em>x\u00a0<\/em>= 1.<\/p>\n<p id=\"fs-id1165135245763\">For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em data-effect=\"italics\">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=\"bcc-box bcc-success\">\n<h3>Try It 9<\/h3>\n<p id=\"fs-id1165137639531\">Solve [latex]5\\mathrm{log}\\left(x+2\\right)=4-\\mathrm{log}\\left(x\\right)\\\\[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135528930\" data-depth=\"2\">\n<h2 data-type=\"title\">Summarizing Translations of the Logarithmic Function<\/h2>\n<p id=\"fs-id1165135528935\">Now that we have worked with each type of translation for the logarithmic function, we can summarize each in the table below\u00a0to arrive at the general equation for translating exponential functions.<\/p>\n<table id=\"Table_04_04_009\" summary=\"Titled,\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Translations of the Parent Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\">Translation<\/th>\n<th style=\"text-align: center;\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift<\/p>\n<ul id=\"fs-id1165137416971\">\n<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\n<li>Vertically <em>d<\/em>\u00a0units up<\/li>\n<\/ul>\n<\/td>\n<td data-align=\"left\">[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 data-align=\"left\">[latex]y=a{\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em data-effect=\"italics\">x<\/em>-axis<\/td>\n<td data-align=\"left\">[latex]y=-{\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td data-align=\"left\">[latex]y={\\mathrm{log}}_{b}\\left(-x\\right)\\\\[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all translations<\/td>\n<td data-align=\"left\">[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d\\\\[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137414493\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Translations of Logarithmic Functions<\/h3>\n<p id=\"fs-id1165137414501\">All translations of the parent logarithmic function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex], have the form<\/p>\n<div id=\"fs-id1165135408512\" class=\"equation\" style=\"text-align: center;\" data-type=\"equation\">[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137734655\">where the parent function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right),b>1\\\\[\/latex], is<\/p>\n<ul id=\"fs-id1165137531610\">\n<li>shifted vertically up <em>d<\/em>\u00a0units.<\/li>\n<li>shifted horizontally to the left <em>c<\/em>\u00a0units.<\/li>\n<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\n<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\n<li>reflected about the <em data-effect=\"italics\">x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<p id=\"fs-id1165137725084\">For [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)\\\\[\/latex], the graph of the parent function is reflected about the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<\/div>\n<div id=\"Example_04_04_10\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135296269\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135296271\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 10: Finding the Vertical Asymptote of a Logarithm Graph<\/h3>\n<p id=\"fs-id1165135296276\">What is the vertical asymptote of [latex]f\\left(x\\right)=-2{\\mathrm{log}}_{3}\\left(x+4\\right)+5\\\\[\/latex]?<\/p>\n<\/div>\n<div id=\"fs-id1165137572550\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137572552\">The vertical asymptote is at <em>x\u00a0<\/em>= \u20134.<\/p>\n<\/div>\n<div id=\"fs-id1165137871955\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165137871960\">The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to <em>x\u00a0<\/em>= \u20134.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 10<\/h3>\n<p id=\"fs-id1165135368433\">What is the vertical asymptote of [latex]f\\left(x\\right)=3+\\mathrm{ln}\\left(x - 1\\right)\\\\[\/latex]?<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<div id=\"Example_04_04_11\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137849555\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137849558\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 11: Finding the Equation from a Graph<\/h3>\n<p id=\"fs-id1165137849563\">Find a possible equation for the common logarithmic function graphed in Figure 15.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201903\/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\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135342977\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165135342979\">This graph has a vertical asymptote at <em>x\u00a0<\/em>= \u20132 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:<\/p>\n<div id=\"eip-id1165133361454\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[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(-1,1\\right)\\\\[\/latex] and [latex]\\left(2,-1\\right)\\\\[\/latex]. Substituting [latex]\\left(-1,1\\right)\\\\[\/latex],<\/p>\n<div id=\"eip-id1165134101923\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}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{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137628655\">Next, substituting in [latex]\\left(2,-1\\right)\\\\[\/latex],<\/p>\n<div id=\"eip-id1165135431720\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}-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{cases}\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135192211\">This gives us the equation [latex]f\\left(x\\right)=-\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1\\\\[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165137735581\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165137735586\">We can verify this answer by comparing the function values in the table below\u00a0with the points on the graph in Example 11.<\/p>\n<table id=\"Table_04_04_010\" summary=\"..\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/><\/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=\"bcc-box bcc-success\">\n<h3>Try It 11<\/h3>\n<p id=\"fs-id1165137665487\">Give the equation of the natural logarithm graphed in Figure 16.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201904\/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\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-20\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<div id=\"fs-id1165137855236\" class=\"note precalculus qa textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"Q&amp;A\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165137855242\"><strong>Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?<\/strong><\/p>\n<p id=\"fs-id1165137827126\"><em data-effect=\"italics\">Yes, if we know the function is a general logarithmic function. For example, look at the graph in Try It 11. The graph approaches x = \u20133 (or thereabouts) more and more closely, so x = \u20133 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex]\\left\\{x|x>-3\\right\\}\\\\[\/latex]. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as [latex]x\\to -{3}^{+},f\\left(x\\right)\\to -\\infty \\\\[\/latex] and as [latex]x\\to \\infty ,f\\left(x\\right)\\to \\infty \\\\[\/latex].<\/em><\/p>\n<\/div>\n<\/section>\n\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-1586\">\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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/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":276,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1586","chapter","type-chapter","status-publish","hentry"],"part":1566,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1586","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1586\/revisions"}],"predecessor-version":[{"id":2326,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1586\/revisions\/2326"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1566"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1586\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/media?parent=1586"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1586"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1586"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/license?post=1586"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}