{"id":2193,"date":"2024-07-11T23:12:37","date_gmt":"2024-07-11T23:12:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2193"},"modified":"2025-08-15T14:33:59","modified_gmt":"2025-08-15T14:33:59","slug":"logarithmic-function-graphs-and-characteristics-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/logarithmic-function-graphs-and-characteristics-apply-it-1\/","title":{"raw":"Logarithmic Function Graphs and Characteristics: Apply It 1","rendered":"Logarithmic Function Graphs and Characteristics: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify the domain of a logarithmic function<\/li>\r\n \t<li>Graph logarithmic functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Transformations of Logarithmic Functions<\/h2>\r\nNow that we have worked with each type of transformation for the logarithmic function, we can summarize each in the table below\u00a0to arrive at the general equation for transforming exponential functions.\r\n<table id=\"Table_04_04_009\" style=\"width: 97.0289%;\" summary=\"Titled,\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 99.2084%;\" colspan=\"2\">Transformations of the Parent Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 65.8311%;\">Translation<\/th>\r\n<th style=\"width: 33.3773%;\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 65.8311%;\">Shift\r\n<ul>\r\n \t<li>Horizontally [latex]c[\/latex]\u00a0units to the left<\/li>\r\n \t<li>Vertically [latex]d[\/latex]\u00a0units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 33.3773%;\">[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 65.8311%;\">Stretch and Compression\r\n<ul>\r\n \t<li>Stretch if [latex]|a|&gt;1[\/latex]<\/li>\r\n \t<li>Compression if [latex]|a|&lt;1[\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 33.3773%;\">[latex]y=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 65.8311%;\">Reflection about the [latex]x[\/latex]-axis<\/td>\r\n<td style=\"width: 33.3773%;\">[latex]y=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 65.8311%;\">Reflection about the [latex]y[\/latex]-axis<\/td>\r\n<td style=\"width: 33.3773%;\">[latex]y={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 65.8311%;\">General equation for all transformations<\/td>\r\n<td style=\"width: 33.3773%;\">[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>transformations of logarithmic functions<\/h3>\r\nAll transformations of the parent logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] have the form\r\n<p style=\"text-align: center;\">[latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/p>\r\nwhere the parent function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right),b&gt;1[\/latex], is\r\n<ul>\r\n \t<li>shifted vertically up [latex]d[\/latex]\u00a0units.<\/li>\r\n \t<li>shifted horizontally to the left [latex]c[\/latex]\u00a0units.<\/li>\r\n \t<li>stretched vertically by a factor of [latex]|a|[\/latex] if [latex]|a| &gt; 0[\/latex].<\/li>\r\n \t<li>compressed vertically by a factor of [latex] |a|[\/latex] if [latex]0 &lt; |a| &lt; 1[\/latex].<\/li>\r\n \t<li>reflected about the [latex]x[\/latex]<em>-<\/em>axis when [latex]a\u00a0&lt; 0[\/latex].<\/li>\r\n<\/ul>\r\nFor [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex], the graph of the parent function is reflected about the [latex]y[\/latex]-axis.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">What is the vertical asymptote of [latex]f\\left(x\\right)=-2{\\mathrm{log}}_{3}\\left(x+4\\right)+5[\/latex]?[reveal-answer q=\"714716\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"714716\"]The vertical asymptote is at [latex]x = \u20134[\/latex]. The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve [latex]4[\/latex] units to the left shifts the vertical asymptote to [latex]x\u00a0= \u20134[\/latex].[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">What is the vertical asymptote of [latex]f\\left(x\\right)=3+\\mathrm{ln}\\left(x - 1\\right)[\/latex]?[reveal-answer q=\"502004\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"502004\"][latex]x=1[\/latex][\/hidden-answer]<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">In the example below, you'll write a common logarithmic function for the graph shown. Remember that all the functions studied in this course possess the characteristic that every point contained on the graph of a function satisfies the equation of the function. As you have done before, begin with the form of a transformed logarithm function, [latex]f(x)=a\\text{log}(x+c)+d[\/latex], then fill in the parts you can discern from the graph.\r\n<ul>\r\n \t<li>Find the horizontal shift by locating the vertical asymptote.<\/li>\r\n \t<li>Examine the shape of the graph to see if it has been reflected.<\/li>\r\n \t<li>Once you have filled in what you know, substitute one or more points in integer coordinates if possible to solve for any remaining unknowns.<\/li>\r\n \t<li>Remember that if there are more than one unknown, you'll need more than one point and more than one equation to solve for all the unknowns.<\/li>\r\n<\/ul>\r\nWork through the example step-by-step with a pencil on paper, perhaps more than once or twice, to gain understanding.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find a possible equation for the common logarithmic function graphed below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233847\/CNX_Precalc_Figure_04_04_021.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).\" width=\"487\" height=\"367\" \/> Graph of f(x)[\/caption]\r\n\r\n[reveal-answer q=\"671912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"671912\"]This graph has a vertical asymptote at [latex]x\u00a0= \u20132[\/latex] and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have the form:\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+k[\/latex]<\/p>\r\nIt appears the graph passes through the points [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(2,-1\\right)[\/latex]. Substituting [latex]\\left(-1,1\\right)[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}1=-a\\mathrm{log}\\left(-1+2\\right)+k\\hfill &amp; \\text{Substitute }\\left(-1,1\\right).\\hfill \\\\ 1=-a\\mathrm{log}\\left(1\\right)+k\\hfill &amp; \\text{Arithmetic}.\\hfill \\\\ 1=k\\hfill &amp; \\text{log(1)}=0.\\hfill \\end{array}[\/latex]<\/p>\r\nNext, substituting [latex]\\left(2,-1\\right)[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllll}-1=-a\\mathrm{log}\\left(2+2\\right)+1\\hfill &amp; \\hfill &amp; \\text{Plug in }\\left(2,-1\\right).\\hfill \\\\ -2=-a\\mathrm{log}\\left(4\\right)\\hfill &amp; \\hfill &amp; \\text{Arithmetic}.\\hfill \\\\ \\text{ }a=\\frac{2}{\\mathrm{log}\\left(4\\right)}\\hfill &amp; \\hfill &amp; \\text{Solve for }a.\\hfill \\end{array}[\/latex]<\/p>\r\nThis gives us the equation [latex]f\\left(x\\right)=-\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1[\/latex].\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nWe can verify this answer by comparing the function values in the table below\u00a0with the points on the graph in this example.\r\n<table id=\"Table_04_04_010\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f(x)[\/latex]<\/strong><\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\u22120.58496[\/latex]<\/td>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]\u22121.3219[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]x<\/strong><\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f(x)[\/latex]<\/strong><\/td>\r\n<td>[latex]\u22121.5850[\/latex]<\/td>\r\n<td>[latex]\u22121.8074[\/latex]<\/td>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]\u22122.1699[\/latex]<\/td>\r\n<td>[latex]\u22122.3219[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Give the equation of the natural logarithm graphed below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233849\/CNX_Precalc_Figure_04_04_022.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).\" width=\"487\" height=\"442\" \/> Graph of f(x)[\/caption]\r\n\r\n[reveal-answer q=\"752379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"752379\"][latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x+3\\right)-1[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24745[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24746[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify the domain of a logarithmic function<\/li>\n<li>Graph logarithmic functions<\/li>\n<\/ul>\n<\/section>\n<h2>Transformations of Logarithmic Functions<\/h2>\n<p>Now that we have worked with each type of transformation for the logarithmic function, we can summarize each in the table below\u00a0to arrive at the general equation for transforming exponential functions.<\/p>\n<table id=\"Table_04_04_009\" style=\"width: 97.0289%;\" summary=\"Titled,\">\n<thead>\n<tr>\n<th style=\"width: 99.2084%;\" colspan=\"2\">Transformations of the Parent Function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<tr>\n<th style=\"width: 65.8311%;\">Translation<\/th>\n<th style=\"width: 33.3773%;\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 65.8311%;\">Shift<\/p>\n<ul>\n<li>Horizontally [latex]c[\/latex]\u00a0units to the left<\/li>\n<li>Vertically [latex]d[\/latex]\u00a0units up<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 33.3773%;\">[latex]y={\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 65.8311%;\">Stretch and Compression<\/p>\n<ul>\n<li>Stretch if [latex]|a|>1[\/latex]<\/li>\n<li>Compression if [latex]|a|<1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 33.3773%;\">[latex]y=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 65.8311%;\">Reflection about the [latex]x[\/latex]-axis<\/td>\n<td style=\"width: 33.3773%;\">[latex]y=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 65.8311%;\">Reflection about the [latex]y[\/latex]-axis<\/td>\n<td style=\"width: 33.3773%;\">[latex]y={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 65.8311%;\">General equation for all transformations<\/td>\n<td style=\"width: 33.3773%;\">[latex]y=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>transformations of logarithmic functions<\/h3>\n<p>All transformations of the parent logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] have the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/p>\n<p>where the parent function, [latex]y={\\mathrm{log}}_{b}\\left(x\\right),b>1[\/latex], is<\/p>\n<ul>\n<li>shifted vertically up [latex]d[\/latex]\u00a0units.<\/li>\n<li>shifted horizontally to the left [latex]c[\/latex]\u00a0units.<\/li>\n<li>stretched vertically by a factor of [latex]|a|[\/latex] if [latex]|a| > 0[\/latex].<\/li>\n<li>compressed vertically by a factor of [latex]|a|[\/latex] if [latex]0 < |a| < 1[\/latex].<\/li>\n<li>reflected about the [latex]x[\/latex]<em>&#8211;<\/em>axis when [latex]a\u00a0< 0[\/latex].<\/li>\n<\/ul>\n<p>For [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex], the graph of the parent function is reflected about the [latex]y[\/latex]-axis.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">What is the vertical asymptote of [latex]f\\left(x\\right)=-2{\\mathrm{log}}_{3}\\left(x+4\\right)+5[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q714716\">Show Solution<\/button><\/p>\n<div id=\"q714716\" class=\"hidden-answer\" style=\"display: none\">The vertical asymptote is at [latex]x = \u20134[\/latex]. The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve [latex]4[\/latex] units to the left shifts the vertical asymptote to [latex]x\u00a0= \u20134[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">What is the vertical asymptote of [latex]f\\left(x\\right)=3+\\mathrm{ln}\\left(x - 1\\right)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q502004\">Show Solution<\/button><\/p>\n<div id=\"q502004\" class=\"hidden-answer\" style=\"display: none\">[latex]x=1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">In the example below, you&#8217;ll write a common logarithmic function for the graph shown. Remember that all the functions studied in this course possess the characteristic that every point contained on the graph of a function satisfies the equation of the function. As you have done before, begin with the form of a transformed logarithm function, [latex]f(x)=a\\text{log}(x+c)+d[\/latex], then fill in the parts you can discern from the graph.<\/p>\n<ul>\n<li>Find the horizontal shift by locating the vertical asymptote.<\/li>\n<li>Examine the shape of the graph to see if it has been reflected.<\/li>\n<li>Once you have filled in what you know, substitute one or more points in integer coordinates if possible to solve for any remaining unknowns.<\/li>\n<li>Remember that if there are more than one unknown, you&#8217;ll need more than one point and more than one equation to solve for all the unknowns.<\/li>\n<\/ul>\n<p>Work through the example step-by-step with a pencil on paper, perhaps more than once or twice, to gain understanding.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find a possible equation for the common logarithmic function graphed below.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233847\/CNX_Precalc_Figure_04_04_021.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).\" width=\"487\" height=\"367\" \/><figcaption class=\"wp-caption-text\">Graph of f(x)<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q671912\">Show Solution<\/button><\/p>\n<div id=\"q671912\" class=\"hidden-answer\" style=\"display: none\">This graph has a vertical asymptote at [latex]x\u00a0= \u20132[\/latex] and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have the form:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-a\\mathrm{log}\\left(x+2\\right)+k[\/latex]<\/p>\n<p>It appears the graph passes through the points [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(2,-1\\right)[\/latex]. Substituting [latex]\\left(-1,1\\right)[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}1=-a\\mathrm{log}\\left(-1+2\\right)+k\\hfill & \\text{Substitute }\\left(-1,1\\right).\\hfill \\\\ 1=-a\\mathrm{log}\\left(1\\right)+k\\hfill & \\text{Arithmetic}.\\hfill \\\\ 1=k\\hfill & \\text{log(1)}=0.\\hfill \\end{array}[\/latex]<\/p>\n<p>Next, substituting [latex]\\left(2,-1\\right)[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllll}-1=-a\\mathrm{log}\\left(2+2\\right)+1\\hfill & \\hfill & \\text{Plug in }\\left(2,-1\\right).\\hfill \\\\ -2=-a\\mathrm{log}\\left(4\\right)\\hfill & \\hfill & \\text{Arithmetic}.\\hfill \\\\ \\text{ }a=\\frac{2}{\\mathrm{log}\\left(4\\right)}\\hfill & \\hfill & \\text{Solve for }a.\\hfill \\end{array}[\/latex]<\/p>\n<p>This gives us the equation [latex]f\\left(x\\right)=-\\frac{2}{\\mathrm{log}\\left(4\\right)}\\mathrm{log}\\left(x+2\\right)+1[\/latex].<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>We can verify this answer by comparing the function values in the table below\u00a0with the points on the graph in this example.<\/p>\n<table id=\"Table_04_04_010\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f(x)[\/latex]<\/strong><\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\u22120.58496[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]\u22121.3219[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]x<\/strong><\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f(x)[\/latex]<\/strong><\/td>\n<td>[latex]\u22121.5850[\/latex]<\/td>\n<td>[latex]\u22121.8074[\/latex]<\/td>\n<td>[latex]\u22122[\/latex]<\/td>\n<td>[latex]\u22122.1699[\/latex]<\/td>\n<td>[latex]\u22122.3219[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Give the equation of the natural logarithm graphed below.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233849\/CNX_Precalc_Figure_04_04_022.jpg\" alt=\"Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\">Graph of f(x)<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q752379\">Show Solution<\/button><\/p>\n<div id=\"q752379\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x+3\\right)-1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24745\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24745&theme=lumen&iframe_resize_id=ohm24745&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24746\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24746&theme=lumen&iframe_resize_id=ohm24746&source=tnh\" width=\"100%\" 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