{"id":1433,"date":"2025-07-25T01:18:07","date_gmt":"2025-07-25T01:18:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1433"},"modified":"2026-03-24T07:42:27","modified_gmt":"2026-03-24T07:42:27","slug":"logarithmic-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-functions-fresh-take\/","title":{"raw":"Logarithmic Functions: Fresh Take","rendered":"Logarithmic Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Convert between logarithmic to exponential form.<\/li>\r\n \t<li>Evaluate logarithms.<\/li>\r\n \t<li>Use common and natural logarithms<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Converting Between Logarithmic And Exponential Form<\/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\">Logarithmic-Exponential Equivalence:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = \\log_b(x)[\/latex] is equivalent to [latex]x = b^y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Logarithms are the inverse of exponential functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Logarithm Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]x &gt; 0, b &gt; 0, b \\neq 1[\/latex]:<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = \\log_b(x)[\/latex] means [latex]b^y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">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\">Domain of logarithm function: [latex](0, \u221e)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range of logarithm function: [latex](-\u221e, \u221e)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Base-10 Convention:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If no base is specified, assume base 10<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Conversion Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Exponential to Logarithmic: Identify base, exponent, and result<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Logarithmic to Exponential: Identify base, argument, and result<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write the following logarithmic equations in exponential form.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex]<\/li>\r\n \t<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"200815\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200815\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex] is equal to [latex]{10}^{6}=1,000,000[\/latex]<\/li>\r\n \t<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex] is equal to [latex]{5}^{2}=25[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write the following exponential equations in logarithmic form.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{3}^{2}=9[\/latex]<\/li>\r\n \t<li>[latex]{5}^{3}=125[\/latex]<\/li>\r\n \t<li>[latex]{2}^{-1}=\\frac{1}{2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"767260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"767260\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{3}^{2}=9[\/latex] is equal to [latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]<\/li>\r\n \t<li>[latex]{5}^{3}=125[\/latex] is equal to [latex]{\\mathrm{log}}_{5}\\left(125\\right)=3[\/latex]<\/li>\r\n \t<li>[latex]{2}^{-1}=\\frac{1}{2}[\/latex] is equal to [latex]{\\text{log}}_{2}\\left(\\frac{1}{2}\\right)=-1[\/latex]<\/li>\r\n<\/ol>\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-hfbageah-z296tOPj0HA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/z296tOPj0HA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hfbageah-z296tOPj0HA\" 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=12850340&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hfbageah-z296tOPj0HA&amp;vembed=0&amp;video_id=z296tOPj0HA&amp;video_target=tpm-plugin-hfbageah-z296tOPj0HA\" 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\/Introduction+to+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Logarithms\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Evaluating Logarithms<\/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\">Mental Evaluation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use knowledge of squares, cubes, and roots to evaluate logarithms mentally<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Remember: \"base to the exponent gives us the number\"<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Logarithm as Exponent:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">In [latex]\\log_b(x) = y[\/latex], [latex]y[\/latex] is the exponent to which [latex]b[\/latex] must be raised to get [latex]x[\/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\">[latex]\\log_b(x) = y[\/latex] is equivalent to [latex]b^y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Negative Exponents:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Remember [latex]b^{-a} = \\frac{1}{b^a}[\/latex] for evaluating logarithms of fractions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex Logarithms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Even seemingly complicated logarithms can often be evaluated mentally by breaking them down<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]y={\\mathrm{log}}_{121}\\left(11\\right)[\/latex] without using a calculator.[reveal-answer q=\"143125\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"143125\"][latex]{\\mathrm{log}}_{121}\\left(11\\right)=\\frac{1}{2}[\/latex] (recall that [latex]\\sqrt{121}={\\left(121\\right)}^{\\frac{1}{2}}=11[\/latex] )[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]y={\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)[\/latex] without using a calculator.[reveal-answer q=\"765423\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"765423\"][latex]{\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)=-5[\/latex][\/hidden-answer]<\/section><section aria-label=\"Example\"><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-ddhchafg-xdBqRQwmlAY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/xdBqRQwmlAY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ddhchafg-xdBqRQwmlAY\" 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=12850341&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ddhchafg-xdBqRQwmlAY&amp;vembed=0&amp;video_id=xdBqRQwmlAY&amp;video_target=tpm-plugin-ddhchafg-xdBqRQwmlAY\" 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\/Evaluating+Basic+Logarithms+Without+a+Calculator_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluating Basic Logarithms Without a Calculator\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section>\r\n<h2>Common Logarithms<\/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\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A common logarithm is a logarithm with base 10<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Written as [latex]\\log(x)[\/latex] or [latex]\\log_{10}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Relationship:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = \\log(x)[\/latex] is equivalent to [latex]10^y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse Function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\log(10^x) = x[\/latex] for all [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]10^{\\log(x)} = x[\/latex] for [latex]x &gt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Richter Scale for earthquakes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Stellar magnitude scale for star brightness<\/li>\r\n \t<li class=\"whitespace-normal break-words\">pH scale for acidity and alkalinity<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Exact values for powers of 10<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Approximate values using calculator for other numbers<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the equation [latex]10^x = 250[\/latex] using common logarithms.[reveal-answer q=\"209108\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"209108\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">To solve this equation, we can take the logarithm (base 10) of both sides: [latex]\\log(10^x) = \\log(250)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using the inverse property of logarithms, we simplify the left side: [latex]x = \\log(250)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Now we need to evaluate [latex]\\log(250)[\/latex]. Since [latex]250[\/latex] is not a power of [latex]10[\/latex], we use a calculator: [latex]x \\approx 2.3979[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">To check our answer, we can use a calculator to compute [latex]10^{2.3979}[\/latex]: [latex]10^{2.3979} \\approx 250.00[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the solution is [latex]x \\approx 2.3979[\/latex].<\/p>\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-hbgfcfbh-Z5myJ8dg_rM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Z5myJ8dg_rM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hbgfcfbh-Z5myJ8dg_rM\" 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=12850342&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hbgfcfbh-Z5myJ8dg_rM&amp;vembed=0&amp;video_id=Z5myJ8dg_rM&amp;video_target=tpm-plugin-hbgfcfbh-Z5myJ8dg_rM\" 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\/Logarithms+%7C+Logarithms+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithms | Logarithms | Algebra II | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n<h2>Natural Logarithms<\/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\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A natural logarithm is a logarithm with base [latex]e[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Written as [latex]\\ln(x)[\/latex] or [latex]\\log_e(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Relationship:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = \\ln(x)[\/latex] is equivalent to [latex]e^y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse Function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\ln(e^x) = x[\/latex] for all [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]e^{\\ln(x)} = x[\/latex] for [latex]x &gt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Properties:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\ln(1) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\ln(e) = 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Most values require a calculator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exception: powers of [latex]e[\/latex] can be evaluated using inverse property<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Natural logarithms are only defined for positive real numbers<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Solve the equation [latex]e^x = 20[\/latex] using natural logarithms.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"363319\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"363319\"]<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">To solve this equation, we can take the natural logarithm of both sides: [latex]\\ln(e^x) = \\ln(20)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using the inverse property of logarithms, we simplify the left side: [latex]x = \\ln(20)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Now we need to evaluate [latex]\\ln(20)[\/latex]. Since [latex]20[\/latex] is not a power of [latex]e[\/latex], we use a calculator: [latex]x \\approx 2.9957[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">To check our answer, we can use a calculator to compute [latex]e^{2.9957}[\/latex]: [latex]e^{2.9957} \\approx 20.00[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the solution is [latex]x \\approx 2.9957[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\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-faedgaae-daUlTsnCNRQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/daUlTsnCNRQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-faedgaae-daUlTsnCNRQ\" 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=12780753&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-faedgaae-daUlTsnCNRQ&amp;vembed=0&amp;video_id=daUlTsnCNRQ&amp;video_target=tpm-plugin-faedgaae-daUlTsnCNRQ\" 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\/Natural+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cNatural Logarithms\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section aria-label=\"Watch It\"><\/section><section aria-label=\"Watch It\"><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-dhhcdgdb-69-QncWjVnw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/69-QncWjVnw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dhhcdgdb-69-QncWjVnw\" 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=12780767&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dhhcdgdb-69-QncWjVnw&amp;vembed=0&amp;video_id=69-QncWjVnw&amp;video_target=tpm-plugin-dhhcdgdb-69-QncWjVnw\" 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\/What+are+natural+logarithms+and+their+properties_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat are natural logarithms and their properties\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Convert between logarithmic to exponential form.<\/li>\n<li>Evaluate logarithms.<\/li>\n<li>Use common and natural logarithms<\/li>\n<\/ul>\n<\/section>\n<h2>Converting Between Logarithmic And Exponential Form<\/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\">Logarithmic-Exponential Equivalence:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]y = \\log_b(x)[\/latex] is equivalent to [latex]x = b^y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Logarithms are the inverse of exponential functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Logarithm Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]x > 0, b > 0, b \\neq 1[\/latex]:<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = \\log_b(x)[\/latex] means [latex]b^y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain and Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain of logarithm function: [latex](0, \u221e)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range of logarithm function: [latex](-\u221e, \u221e)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Base-10 Convention:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If no base is specified, assume base 10<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conversion Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exponential to Logarithmic: Identify base, exponent, and result<\/li>\n<li class=\"whitespace-normal break-words\">Logarithmic to Exponential: Identify base, argument, and result<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write the following logarithmic equations in exponential form.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex]<\/li>\n<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q200815\">Show Solution<\/button><\/p>\n<div id=\"q200815\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex] is equal to [latex]{10}^{6}=1,000,000[\/latex]<\/li>\n<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex] is equal to [latex]{5}^{2}=25[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the following exponential equations in logarithmic form.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{3}^{2}=9[\/latex]<\/li>\n<li>[latex]{5}^{3}=125[\/latex]<\/li>\n<li>[latex]{2}^{-1}=\\frac{1}{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q767260\">Show Solution<\/button><\/p>\n<div id=\"q767260\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{3}^{2}=9[\/latex] is equal to [latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]<\/li>\n<li>[latex]{5}^{3}=125[\/latex] is equal to [latex]{\\mathrm{log}}_{5}\\left(125\\right)=3[\/latex]<\/li>\n<li>[latex]{2}^{-1}=\\frac{1}{2}[\/latex] is equal to [latex]{\\text{log}}_{2}\\left(\\frac{1}{2}\\right)=-1[\/latex]<\/li>\n<\/ol>\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-hfbageah-z296tOPj0HA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/z296tOPj0HA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hfbageah-z296tOPj0HA\" 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=12850340&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hfbageah-z296tOPj0HA&amp;vembed=0&amp;video_id=z296tOPj0HA&amp;video_target=tpm-plugin-hfbageah-z296tOPj0HA\" 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\/Introduction+to+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Logarithms\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Evaluating Logarithms<\/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\">Mental Evaluation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use knowledge of squares, cubes, and roots to evaluate logarithms mentally<\/li>\n<li class=\"whitespace-normal break-words\">Remember: &#8220;base to the exponent gives us the number&#8221;<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Logarithm as Exponent:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">In [latex]\\log_b(x) = y[\/latex], [latex]y[\/latex] is the exponent to which [latex]b[\/latex] must be raised to get [latex]x[\/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\">[latex]\\log_b(x) = y[\/latex] is equivalent to [latex]b^y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Negative Exponents:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Remember [latex]b^{-a} = \\frac{1}{b^a}[\/latex] for evaluating logarithms of fractions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Complex Logarithms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Even seemingly complicated logarithms can often be evaluated mentally by breaking them down<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]y={\\mathrm{log}}_{121}\\left(11\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q143125\">Show Solution<\/button><\/p>\n<div id=\"q143125\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\mathrm{log}}_{121}\\left(11\\right)=\\frac{1}{2}[\/latex] (recall that [latex]\\sqrt{121}={\\left(121\\right)}^{\\frac{1}{2}}=11[\/latex] )<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]y={\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q765423\">Show Solution<\/button><\/p>\n<div id=\"q765423\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)=-5[\/latex]<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\">\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-ddhchafg-xdBqRQwmlAY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/xdBqRQwmlAY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ddhchafg-xdBqRQwmlAY\" 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=12850341&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ddhchafg-xdBqRQwmlAY&amp;vembed=0&amp;video_id=xdBqRQwmlAY&amp;video_target=tpm-plugin-ddhchafg-xdBqRQwmlAY\" 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\/Evaluating+Basic+Logarithms+Without+a+Calculator_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluating Basic Logarithms Without a Calculator\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n<h2>Common Logarithms<\/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\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A common logarithm is a logarithm with base 10<\/li>\n<li class=\"whitespace-normal break-words\">Written as [latex]\\log(x)[\/latex] or [latex]\\log_{10}(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]y = \\log(x)[\/latex] is equivalent to [latex]10^y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log(10^x) = x[\/latex] for all [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]10^{\\log(x)} = x[\/latex] for [latex]x > 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Richter Scale for earthquakes<\/li>\n<li class=\"whitespace-normal break-words\">Stellar magnitude scale for star brightness<\/li>\n<li class=\"whitespace-normal break-words\">pH scale for acidity and alkalinity<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exact values for powers of 10<\/li>\n<li class=\"whitespace-normal break-words\">Approximate values using calculator for other numbers<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the equation [latex]10^x = 250[\/latex] using common logarithms.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q209108\">Show Answer<\/button><\/p>\n<div id=\"q209108\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">To solve this equation, we can take the logarithm (base 10) of both sides: [latex]\\log(10^x) = \\log(250)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Using the inverse property of logarithms, we simplify the left side: [latex]x = \\log(250)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Now we need to evaluate [latex]\\log(250)[\/latex]. Since [latex]250[\/latex] is not a power of [latex]10[\/latex], we use a calculator: [latex]x \\approx 2.3979[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">To check our answer, we can use a calculator to compute [latex]10^{2.3979}[\/latex]: [latex]10^{2.3979} \\approx 250.00[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the solution is [latex]x \\approx 2.3979[\/latex].<\/p>\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-hbgfcfbh-Z5myJ8dg_rM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Z5myJ8dg_rM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hbgfcfbh-Z5myJ8dg_rM\" 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=12850342&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hbgfcfbh-Z5myJ8dg_rM&amp;vembed=0&amp;video_id=Z5myJ8dg_rM&amp;video_target=tpm-plugin-hbgfcfbh-Z5myJ8dg_rM\" 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\/Logarithms+%7C+Logarithms+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithms | Logarithms | Algebra II | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<h2>Natural Logarithms<\/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\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A natural logarithm is a logarithm with base [latex]e[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Written as [latex]\\ln(x)[\/latex] or [latex]\\log_e(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]y = \\ln(x)[\/latex] is equivalent to [latex]e^y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\ln(e^x) = x[\/latex] for all [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e^{\\ln(x)} = x[\/latex] for [latex]x > 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\ln(1) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\ln(e) = 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Most values require a calculator<\/li>\n<li class=\"whitespace-normal break-words\">Exception: powers of [latex]e[\/latex] can be evaluated using inverse property<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Natural logarithms are only defined for positive real numbers<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Solve the equation [latex]e^x = 20[\/latex] using natural logarithms.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q363319\">Show Answer<\/button><\/p>\n<div id=\"q363319\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">To solve this equation, we can take the natural logarithm of both sides: [latex]\\ln(e^x) = \\ln(20)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Using the inverse property of logarithms, we simplify the left side: [latex]x = \\ln(20)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Now we need to evaluate [latex]\\ln(20)[\/latex]. Since [latex]20[\/latex] is not a power of [latex]e[\/latex], we use a calculator: [latex]x \\approx 2.9957[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">To check our answer, we can use a calculator to compute [latex]e^{2.9957}[\/latex]: [latex]e^{2.9957} \\approx 20.00[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the solution is [latex]x \\approx 2.9957[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/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-faedgaae-daUlTsnCNRQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/daUlTsnCNRQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-faedgaae-daUlTsnCNRQ\" 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=12780753&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-faedgaae-daUlTsnCNRQ&amp;vembed=0&amp;video_id=daUlTsnCNRQ&amp;video_target=tpm-plugin-faedgaae-daUlTsnCNRQ\" 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\/Natural+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cNatural Logarithms\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section aria-label=\"Watch It\"><\/section>\n<section aria-label=\"Watch It\">\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-dhhcdgdb-69-QncWjVnw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/69-QncWjVnw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dhhcdgdb-69-QncWjVnw\" 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=12780767&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dhhcdgdb-69-QncWjVnw&amp;vembed=0&amp;video_id=69-QncWjVnw&amp;video_target=tpm-plugin-dhhcdgdb-69-QncWjVnw\" 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\/What+are+natural+logarithms+and+their+properties_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat are natural logarithms and their properties\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n","protected":false},"author":67,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Introduction to Logarithms\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/z296tOPj0HA\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Evaluating Basic Logarithms Without a Calculator\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/xdBqRQwmlAY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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