{"id":1436,"date":"2025-07-25T01:20:19","date_gmt":"2025-07-25T01:20:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1436"},"modified":"2026-03-24T07:40:28","modified_gmt":"2026-03-24T07:40:28","slug":"graphs-of-logarithmic-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-logarithmic-functions-fresh-take\/","title":{"raw":"Graphs of Logarithmic Functions: Fresh Take","rendered":"Graphs of Logarithmic Functions: Fresh Take"},"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>Domain of Logarithmic Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Basic Domain Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]y = \\log_b(x)[\/latex], the domain is [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The argument of a logarithm must be positive<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Asymptote:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Logarithmic functions have a vertical asymptote at [latex]x = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse Relationship:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain of [latex]\\log_b(x)[\/latex] is the range of [latex]b^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range of [latex]\\log_b(x)[\/latex] is the domain of [latex]b^x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Transformations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Can change the domain of the parent function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Always ensure the argument remains positive<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Finding Domains:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set up inequality: [latex]\\text{ argument }&gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Express domain in interval notation<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x - 2\\right)+1[\/latex]?[reveal-answer q=\"613113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"613113\"][latex]\\left(2,\\infty \\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(x - 5\\right)+2[\/latex]?[reveal-answer q=\"983551\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"983551\"][latex]\\left(5,\\infty \\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gfehebda-_Om0ZMzIgUk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_Om0ZMzIgUk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gfehebda-_Om0ZMzIgUk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850344&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gfehebda-_Om0ZMzIgUk&amp;vembed=0&amp;video_id=_Om0ZMzIgUk&amp;video_target=tpm-plugin-gfehebda-_Om0ZMzIgUk\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+the+Domain+of+Logarithmic+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Domain of Logarithmic Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Graphing a Logarithmic Function Using a Table of Values<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Parent Function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = \\log_b(x)[\/latex] where [latex]b &gt; 0[\/latex] and [latex]b \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse Relationship:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Logarithmic functions are inverses of exponential functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Their graphs are reflections of each other across [latex]y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptote: [latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercept: [latex](1, 0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key point: [latex](b, 1)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Behavior:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Increasing if [latex]b &gt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Decreasing if [latex]0 &lt; b &lt; 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph Shape:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Starts at vertical asymptote [latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Passes through [latex](1, 0)[\/latex] and [latex](b, 1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Curves upward ([latex]b &gt; 1[\/latex]) or downward ([latex]0 &lt; b &lt; 1[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdhaffag-w1A2ZYmfGco\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/w1A2ZYmfGco?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cdhaffag-w1A2ZYmfGco\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850345&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cdhaffag-w1A2ZYmfGco&amp;vembed=0&amp;video_id=w1A2ZYmfGco&amp;video_target=tpm-plugin-cdhaffag-w1A2ZYmfGco\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Graph+an+Exponential+Function+and+Logarithmic+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Graph an Exponential Function and Logarithmic Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bfbdaaha-GnfclmCE9rE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/GnfclmCE9rE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bfbdaaha-GnfclmCE9rE\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850416&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bfbdaaha-GnfclmCE9rE&amp;vembed=0&amp;video_id=GnfclmCE9rE&amp;video_target=tpm-plugin-bfbdaaha-GnfclmCE9rE\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Match+Graphs+with+Exponential+and+Logarithmic+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Match Graphs with Exponential and Logarithmic Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Graphing Transformations of Logarithmic Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Parent Function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = \\log_b(x)[\/latex] where [latex]b &gt; 0[\/latex] and [latex]b \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Shifts:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = \\log_b(x + c)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shifts left [latex]c[\/latex] units if [latex]c &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shifts right [latex]c[\/latex] units if [latex]c &lt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptote: [latex]x = -c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex](-c, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Shifts:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = \\log_b(x) + d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shifts up [latex]d[\/latex] units if [latex]d &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shifts down [latex]d[\/latex] units if [latex]d &lt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptote: [latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effect on Domain and Range:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal shifts affect domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical shifts do not affect domain or range<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Stretches and Compressions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = a\\log_b(x)[\/latex], where [latex]a &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Stretch if [latex]a &gt; 1[\/latex], compress if [latex]0 &lt; a &lt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptote: [latex]x = 0[\/latex] (unchanged)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex](0, \\infty)[\/latex] (unchanged)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex](-\\infty, \\infty)[\/latex] (unchanged)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reflections:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">About [latex]x[\/latex]-axis: [latex]f(x) = -\\log_b(x)[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex](0, \\infty)[\/latex], Range: [latex](-\\infty, \\infty)[\/latex] (unchanged)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">About [latex]y[\/latex]-axis: [latex]f(x) = \\log_b(-x)[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex](-\\infty, 0)[\/latex], Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combined Transformations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Can involve multiple operations (e.g., [latex]f(x) = a\\log_b(x-h) + k[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply transformations in the correct order: inside parentheses first, then outside<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Asymptote:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x = 0[\/latex] for parent function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shifts with horizontal transformations<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">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.[reveal-answer q=\"779370\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"779370\"]The domain is [latex]\\left(-4,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the asymptote <em>x\u00a0<\/em>= \u20134.\r\n\r\n[caption id=\"attachment_3106\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3106 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16230941\/CNX_Precalc_Figure_04_04_0092.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).\" width=\"487\" height=\"363\" \/> Graph of a parent function and its translation, f(x)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">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.[reveal-answer q=\"338440\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338440\"]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.\r\n\r\n[caption id=\"attachment_3109\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3109 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16231838\/CNX_Precalc_Figure_04_04_0122.jpg\" alt=\"Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).\" width=\"487\" height=\"474\" \/> Graph of a parent function and its translation, f(x)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">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.[reveal-answer q=\"250125\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"250125\"]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.\r\n\r\n[caption id=\"attachment_3114\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3114 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16233042\/CNX_Precalc_Figure_04_04_0152.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1\/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).\" width=\"487\" height=\"364\" \/> Graph of a parent function and its translation, f(x)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">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.[reveal-answer q=\"404704\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"404704\"]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.\r\n\r\n[caption id=\"attachment_3115\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3115 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16233717\/CNX_Precalc_Figure_04_04_0172.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" width=\"487\" height=\"439\" \/> Graph of f(x) with an asymptote at x=2[\/caption]\r\n\r\n<div id=\"fs-id1165137437228\" class=\"solution\"><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right)[\/latex]. State the domain, range, and asymptote.[reveal-answer q=\"160849\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160849\"]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.\r\n\r\n[caption id=\"attachment_3117\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3117 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16234244\/CNX_Precalc_Figure_04_04_0202.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" width=\"487\" height=\"288\" \/> Graph of f(x) with an asymptote at x=0[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-efhadcgb-OnhJcrYZmEo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/OnhJcrYZmEo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-efhadcgb-OnhJcrYZmEo\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780768&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-efhadcgb-OnhJcrYZmEo&amp;vembed=0&amp;video_id=OnhJcrYZmEo&amp;video_target=tpm-plugin-efhadcgb-OnhJcrYZmEo\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+graph+a+logarithmic+function+with+horizontal+shift_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to graph a logarithmic function with horizontal shift\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hffddbbc-TpQXDnCzsS8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/TpQXDnCzsS8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hffddbbc-TpQXDnCzsS8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780770&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hffddbbc-TpQXDnCzsS8&amp;vembed=0&amp;video_id=TpQXDnCzsS8&amp;video_target=tpm-plugin-hffddbbc-TpQXDnCzsS8\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+logarithmic+function+with+multiple+transformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a logarithmic function with multiple transformations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbhdfhbb-b6TRv3oJ0ZI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/b6TRv3oJ0ZI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbhdfhbb-b6TRv3oJ0ZI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780771&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbhdfhbb-b6TRv3oJ0ZI&amp;vembed=0&amp;video_id=b6TRv3oJ0ZI&amp;video_target=tpm-plugin-fbhdfhbb-b6TRv3oJ0ZI\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Logarithmic+Function+Graph+Stretch+and+Compression+Shifts%3ATransformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithmic Function Graph Stretch and Compression Shifts\/Transformations\" here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbdchgef-VOAi_CkdlCQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/VOAi_CkdlCQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbdchgef-VOAi_CkdlCQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780772&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbdchgef-VOAi_CkdlCQ&amp;vembed=0&amp;video_id=VOAi_CkdlCQ&amp;video_target=tpm-plugin-bbdchgef-VOAi_CkdlCQ\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Learn+how+to+graph+a+logarithm+with+reflections+over+x+and+y+axis_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLearn how to graph a logarithm with reflections over x and y axis\u201d here (opens in new window).<\/a>\r\n\r\n<\/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>Domain of Logarithmic Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Basic Domain Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]y = \\log_b(x)[\/latex], the domain is [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The argument of a logarithm must be positive<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Asymptote:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Logarithmic functions have a vertical asymptote at [latex]x = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain of [latex]\\log_b(x)[\/latex] is the range of [latex]b^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range of [latex]\\log_b(x)[\/latex] is the domain of [latex]b^x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Can change the domain of the parent function<\/li>\n<li class=\"whitespace-normal break-words\">Always ensure the argument remains positive<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Finding Domains:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set up inequality: [latex]\\text{ argument }> 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Express domain in interval notation<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x - 2\\right)+1[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q613113\">Show Solution<\/button><\/p>\n<div id=\"q613113\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(2,\\infty \\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(x - 5\\right)+2[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q983551\">Show Solution<\/button><\/p>\n<div id=\"q983551\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(5,\\infty \\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gfehebda-_Om0ZMzIgUk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_Om0ZMzIgUk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gfehebda-_Om0ZMzIgUk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850344&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gfehebda-_Om0ZMzIgUk&amp;vembed=0&amp;video_id=_Om0ZMzIgUk&amp;video_target=tpm-plugin-gfehebda-_Om0ZMzIgUk\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+the+Domain+of+Logarithmic+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Domain of Logarithmic Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Graphing a Logarithmic Function Using a Table of Values<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Parent Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = \\log_b(x)[\/latex] where [latex]b > 0[\/latex] and [latex]b \\neq 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Logarithmic functions are inverses of exponential functions<\/li>\n<li class=\"whitespace-normal break-words\">Their graphs are reflections of each other across [latex]y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical asymptote: [latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercept: [latex](1, 0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key point: [latex](b, 1)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Increasing if [latex]b > 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Decreasing if [latex]0 < b < 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graph Shape:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Starts at vertical asymptote [latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Passes through [latex](1, 0)[\/latex] and [latex](b, 1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Curves upward ([latex]b > 1[\/latex]) or downward ([latex]0 < b < 1[\/latex])<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdhaffag-w1A2ZYmfGco\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/w1A2ZYmfGco?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cdhaffag-w1A2ZYmfGco\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850345&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cdhaffag-w1A2ZYmfGco&amp;vembed=0&amp;video_id=w1A2ZYmfGco&amp;video_target=tpm-plugin-cdhaffag-w1A2ZYmfGco\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Graph+an+Exponential+Function+and+Logarithmic+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Graph an Exponential Function and Logarithmic Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bfbdaaha-GnfclmCE9rE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/GnfclmCE9rE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bfbdaaha-GnfclmCE9rE\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850416&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bfbdaaha-GnfclmCE9rE&amp;vembed=0&amp;video_id=GnfclmCE9rE&amp;video_target=tpm-plugin-bfbdaaha-GnfclmCE9rE\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Match+Graphs+with+Exponential+and+Logarithmic+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Match Graphs with Exponential and Logarithmic Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Graphing Transformations of Logarithmic Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Parent Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = \\log_b(x)[\/latex] where [latex]b > 0[\/latex] and [latex]b \\neq 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Shifts:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = \\log_b(x + c)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shifts left [latex]c[\/latex] units if [latex]c > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shifts right [latex]c[\/latex] units if [latex]c < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical asymptote: [latex]x = -c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Domain: [latex](-c, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Shifts:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = \\log_b(x) + d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shifts up [latex]d[\/latex] units if [latex]d > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shifts down [latex]d[\/latex] units if [latex]d < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical asymptote: [latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Domain: [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effect on Domain and Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal shifts affect domain<\/li>\n<li class=\"whitespace-normal break-words\">Vertical shifts do not affect domain or range<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Stretches and Compressions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = a\\log_b(x)[\/latex], where [latex]a > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Stretch if [latex]a > 1[\/latex], compress if [latex]0 < a < 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical asymptote: [latex]x = 0[\/latex] (unchanged)<\/li>\n<li class=\"whitespace-normal break-words\">Domain: [latex](0, \\infty)[\/latex] (unchanged)<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex](-\\infty, \\infty)[\/latex] (unchanged)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reflections:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">About [latex]x[\/latex]-axis: [latex]f(x) = -\\log_b(x)[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: [latex](0, \\infty)[\/latex], Range: [latex](-\\infty, \\infty)[\/latex] (unchanged)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">About [latex]y[\/latex]-axis: [latex]f(x) = \\log_b(-x)[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: [latex](-\\infty, 0)[\/latex], Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Combined Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Can involve multiple operations (e.g., [latex]f(x) = a\\log_b(x-h) + k[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Apply transformations in the correct order: inside parentheses first, then outside<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Asymptote:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x = 0[\/latex] for parent function<\/li>\n<li class=\"whitespace-normal break-words\">Shifts with horizontal transformations<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q779370\">Show Solution<\/button><\/p>\n<div id=\"q779370\" class=\"hidden-answer\" style=\"display: none\">The domain is [latex]\\left(-4,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the asymptote <em>x\u00a0<\/em>= \u20134.<\/p>\n<figure id=\"attachment_3106\" aria-describedby=\"caption-attachment-3106\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3106 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16230941\/CNX_Precalc_Figure_04_04_0092.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).\" width=\"487\" height=\"363\" \/><figcaption id=\"caption-attachment-3106\" class=\"wp-caption-text\">Graph of a parent function and its translation, f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)+2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q338440\">Show Solution<\/button><\/p>\n<div id=\"q338440\" class=\"hidden-answer\" style=\"display: none\">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<figure id=\"attachment_3109\" aria-describedby=\"caption-attachment-3109\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3109 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16231838\/CNX_Precalc_Figure_04_04_0122.jpg\" alt=\"Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).\" width=\"487\" height=\"474\" \/><figcaption id=\"caption-attachment-3109\" class=\"wp-caption-text\">Graph of a parent function and its translation, f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q250125\">Show Solution<\/button><\/p>\n<div id=\"q250125\" class=\"hidden-answer\" style=\"display: none\">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<figure id=\"attachment_3114\" aria-describedby=\"caption-attachment-3114\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3114 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16233042\/CNX_Precalc_Figure_04_04_0152.jpg\" alt=\"Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1\/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).\" width=\"487\" height=\"364\" \/><figcaption id=\"caption-attachment-3114\" class=\"wp-caption-text\">Graph of a parent function and its translation, f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch a graph of the function [latex]f\\left(x\\right)=3\\mathrm{log}\\left(x - 2\\right)+1[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q404704\">Show Solution<\/button><\/p>\n<div id=\"q404704\" class=\"hidden-answer\" style=\"display: none\">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<figure id=\"attachment_3115\" aria-describedby=\"caption-attachment-3115\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3115 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16233717\/CNX_Precalc_Figure_04_04_0172.jpg\" alt=\"Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.\" width=\"487\" height=\"439\" \/><figcaption id=\"caption-attachment-3115\" class=\"wp-caption-text\">Graph of f(x) with an asymptote at x=2<\/figcaption><\/figure>\n<div id=\"fs-id1165137437228\" class=\"solution\"><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph [latex]f\\left(x\\right)=-\\mathrm{log}\\left(-x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q160849\">Show Solution<\/button><\/p>\n<div id=\"q160849\" class=\"hidden-answer\" style=\"display: none\">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<figure id=\"attachment_3117\" aria-describedby=\"caption-attachment-3117\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3117 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16234244\/CNX_Precalc_Figure_04_04_0202.jpg\" alt=\"Graph of f(x)=-log(-x) with an asymptote at x=0.\" width=\"487\" height=\"288\" \/><figcaption id=\"caption-attachment-3117\" class=\"wp-caption-text\">Graph of f(x) with an asymptote at x=0<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-efhadcgb-OnhJcrYZmEo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/OnhJcrYZmEo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-efhadcgb-OnhJcrYZmEo\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780768&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-efhadcgb-OnhJcrYZmEo&amp;vembed=0&amp;video_id=OnhJcrYZmEo&amp;video_target=tpm-plugin-efhadcgb-OnhJcrYZmEo\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+graph+a+logarithmic+function+with+horizontal+shift_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to graph a logarithmic function with horizontal shift\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hffddbbc-TpQXDnCzsS8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/TpQXDnCzsS8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hffddbbc-TpQXDnCzsS8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780770&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hffddbbc-TpQXDnCzsS8&amp;vembed=0&amp;video_id=TpQXDnCzsS8&amp;video_target=tpm-plugin-hffddbbc-TpQXDnCzsS8\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+logarithmic+function+with+multiple+transformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a logarithmic function with multiple transformations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbhdfhbb-b6TRv3oJ0ZI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/b6TRv3oJ0ZI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbhdfhbb-b6TRv3oJ0ZI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780771&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbhdfhbb-b6TRv3oJ0ZI&amp;vembed=0&amp;video_id=b6TRv3oJ0ZI&amp;video_target=tpm-plugin-fbhdfhbb-b6TRv3oJ0ZI\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Logarithmic+Function+Graph+Stretch+and+Compression+Shifts%3ATransformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithmic Function Graph Stretch and Compression Shifts\/Transformations&#8221; here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbdchgef-VOAi_CkdlCQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/VOAi_CkdlCQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbdchgef-VOAi_CkdlCQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780772&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbdchgef-VOAi_CkdlCQ&amp;vembed=0&amp;video_id=VOAi_CkdlCQ&amp;video_target=tpm-plugin-bbdchgef-VOAi_CkdlCQ\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Learn+how+to+graph+a+logarithm+with+reflections+over+x+and+y+axis_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLearn how to graph a logarithm with reflections over x and y axis\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":29,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Find the Domain of Logarithmic Functions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/_Om0ZMzIgUk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Graph an Exponential Function and Logarithmic Function\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/w1A2ZYmfGco\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Match Graphs with Exponential and Logarithmic Functions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/GnfclmCE9rE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How to graph a logarithmic function with horizontal shift\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/OnhJcrYZmEo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Graph a logarithmic function with multiple transformations\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/TpQXDnCzsS8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Logarithmic Function Graph Stretch and Compression Shifts\/Transformations\",\"author\":\"\",\"organization\":\"Math Tutorials\",\"url\":\"https:\/\/youtu.be\/b6TRv3oJ0ZI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Learn how to graph a logarithm with reflections over x and y axis\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/VOAi_CkdlCQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":105,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Ex: Find the Domain of Logarithmic Functions","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/_Om0ZMzIgUk","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex: Graph an Exponential Function and Logarithmic Function","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/w1A2ZYmfGco","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 1: Match Graphs with Exponential and Logarithmic Functions","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/GnfclmCE9rE","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"How to graph a logarithmic function with horizontal shift","author":"Brian McLogan","organization":"","url":"https:\/\/youtu.be\/OnhJcrYZmEo","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Graph a logarithmic function with multiple transformations","author":"Brian McLogan","organization":"","url":"https:\/\/youtu.be\/TpQXDnCzsS8","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Logarithmic Function Graph Stretch and Compression Shifts\/Transformations","author":"","organization":"Math Tutorials","url":"https:\/\/youtu.be\/b6TRv3oJ0ZI","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Learn how to graph a logarithm with reflections over x and y axis","author":"Brian 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