{"id":1440,"date":"2025-07-25T01:26:31","date_gmt":"2025-07-25T01:26:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1440"},"modified":"2026-03-24T07:36:39","modified_gmt":"2026-03-24T07:36:39","slug":"properties-of-logarithms-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/properties-of-logarithms-fresh-take\/","title":{"raw":"Logarithmic Properties: Fresh Take","rendered":"Logarithmic Properties: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Expand logarithmic expressions.<\/li>\r\n \t<li>Condense logarithmic expressions.<\/li>\r\n \t<li>Use the change-of-base formula for logarithms.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Basic Properties of 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\">Logarithmic-Exponential 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 \\Leftrightarrow b^y = x[\/latex], where [latex]b &gt; 0, b \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Zero Property:\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(1) = 0[\/latex] for any base [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equivalent to [latex]b^0 = 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identity Property:\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(b) = 1[\/latex] for any base [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equivalent to [latex]b^1 = b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse Properties:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]b^{\\log_b(x)} = x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\log_b(b^x) = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain Restrictions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]b &gt; 0, b \\neq 1[\/latex] (base restriction)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x &gt; 0[\/latex] (argument restriction)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Using the Product Rule for 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\">Product Rule Statement:\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(MN) = \\log_b(M) + \\log_b(N)[\/latex], where [latex]b &gt; 0, b \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Relation to Exponent Rules:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Analogous to [latex]x^a \\cdot x^b = x^{a+b}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiple Factors:\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(wxyz) = \\log_b(w) + \\log_b(x) + \\log_b(y) + \\log_b(z)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex]{\\mathrm{log}}_{b}\\left(8k\\right)[\/latex].[reveal-answer q=\"829261\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"829261\"][latex]{\\mathrm{log}}_{b}8+{\\mathrm{log}}_{b}k[\/latex][\/hidden-answer]<\/section>\r\n<h2>Using the Quotient Rule for 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\">Quotient Rule Statement:\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(\\frac{M}{N}) = \\log_b(M) - \\log_b(N)[\/latex], where [latex]b &gt; 0, b \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Relation to Exponent Rules:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Analogous to [latex]\\frac{x^a}{x^b} = x^{a-b}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combination with Product Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Often used in conjunction with the product rule for full expansion<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex]{\\mathrm{log}}_{3}\\left(\\frac{7{x}^{2}+21x}{7x\\left(x - 1\\right)\\left(x - 2\\right)}\\right)[\/latex].[reveal-answer q=\"442008\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"442008\"][latex]{\\mathrm{log}}_{3}\\left(x+3\\right)-{\\mathrm{log}}_{3}\\left(x - 1\\right)-{\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex][\/hidden-answer]<\/section>\r\n<h2>Using the Power Rule for 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\">Power Rule Statement:\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(M^n) = n \\log_b(M)[\/latex], where [latex]b &gt; 0, b \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Derivation from Product Rule:\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^2) = \\log_b(x \\cdot x) = \\log_b(x) + \\log_b(x) = 2\\log_b(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Application to Roots:\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(\\sqrt{x}) = \\log_b(x^{\\frac{1}{2}}) = \\frac{1}{2}\\log_b(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\mathrm{ln}{x}^{2}[\/latex].[reveal-answer q=\"383972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"383972\"][latex]2\\mathrm{ln}x[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\mathrm{ln}\\left(\\frac{1}{{x}^{2}}\\right)[\/latex].[reveal-answer q=\"947582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"947582\"][latex]-2\\mathrm{ln}\\left(x\\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-fbebcceb-SxF44olWTyk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SxF44olWTyk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbebcceb-SxF44olWTyk\" 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=12850417&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbebcceb-SxF44olWTyk&amp;vembed=0&amp;video_id=SxF44olWTyk&amp;video_target=tpm-plugin-fbebcceb-SxF44olWTyk\" 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\/The+Properties+of+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Properties of Logarithms\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-afefbgch-OiVILGg227E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/OiVILGg227E?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-afefbgch-OiVILGg227E\" 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=12850418&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-afefbgch-OiVILGg227E&amp;vembed=0&amp;video_id=OiVILGg227E&amp;video_target=tpm-plugin-afefbgch-OiVILGg227E\" 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+-+Evaluate+a+Natural+Logarithmic+Expression+Using+the+Properties+of+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Evaluate a Natural Logarithmic Expression Using the Properties of Logarithms\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Expanding and Condensing 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\">Expanding Logarithms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Apply rules to break down complex logarithmic expressions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can only expand products, quotients, powers, and roots inside the logarithm<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Condensing Logarithms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Combine multiple logarithms with the same base into a single logarithm<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply rules in reverse order: Power rule first, then product\/quotient rules<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Important Considerations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Logarithms must have the same base to be combined<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Addition or subtraction inside the logarithm cannot be expanded<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>Tip for success<\/strong>\r\n<ul>\r\n \t<li>When expanding and condensing logarithms, keep in mind that there are often more than one or two good ways to reach a good conclusion. The rules for manipulating exponents and logarithms can be combined creatively. You should try a few different ideas for using the rules on complicated expressions to get practice for finding the most efficient path to take in different situations.<\/li>\r\n \t<li>When working with more complicated examples, write your work down step by step to avoid making incorrect assumptions. It's okay to try different rules as you practice creativity as long as you use each rule correctly.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex]\\mathrm{log}\\left(\\frac{{x}^{2}{y}^{3}}{{z}^{4}}\\right)[\/latex].[reveal-answer q=\"722800\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"722800\"][latex]2\\mathrm{log}x+3\\mathrm{log}y - 4\\mathrm{log}z[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Expand [latex]\\mathrm{ln}\\left(\\sqrt[3]{{x}^{2}}\\right)[\/latex].[reveal-answer q=\"2296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"2296\"][latex]\\frac{2}{3}\\mathrm{ln}x[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Expand [latex]\\mathrm{ln}\\left(\\frac{\\sqrt{\\left(x - 1\\right){\\left(2x+1\\right)}^{2}}}{\\left({x}^{2}-9\\right)}\\right)[\/latex].[reveal-answer q=\"767425\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"767425\"][latex]\\frac{1}{2}\\mathrm{ln}\\left(x - 1\\right)+\\mathrm{ln}\\left(2x+1\\right)-\\mathrm{ln}\\left(x+3\\right)-\\mathrm{ln}\\left(x - 3\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Use the power rule for logs to rewrite [latex]2{\\mathrm{log}}_{3}4[\/latex] as a single logarithm with a leading coefficient of 1.[reveal-answer q=\"720709\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"720709\"][latex]{\\mathrm{log}}_{3}16[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Condense [latex]\\mathrm{log}3-\\mathrm{log}4+\\mathrm{log}5-\\mathrm{log}6[\/latex].[reveal-answer q=\"52020\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"52020\"][latex]\\mathrm{log}\\left(\\frac{3\\cdot 5}{4\\cdot 6}\\right)[\/latex]; can also be written [latex]\\mathrm{log}\\left(\\frac{5}{8}\\right)[\/latex] by simplifying the fraction to lowest terms.[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\mathrm{log}\\left(5\\right)+0.5\\mathrm{log}\\left(x\\right)-\\mathrm{log}\\left(7x - 1\\right)+3\\mathrm{log}\\left(x - 1\\right)[\/latex] as a single logarithm.[reveal-answer q=\"812624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"812624\"][latex]\\mathrm{log}\\left(\\frac{5{\\left(x - 1\\right)}^{3}\\sqrt{x}}{\\left(7x - 1\\right)}\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Condense [latex]4\\left(3\\mathrm{log}\\left(x\\right)+\\mathrm{log}\\left(x+5\\right)-\\mathrm{log}\\left(2x+3\\right)\\right)[\/latex].[reveal-answer q=\"622494\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"622494\"][latex]\\mathrm{log}\\frac{{x}^{12}{\\left(x+5\\right)}^{4}}{{\\left(2x+3\\right)}^{4}}[\/latex]; this answer could also be written as [latex]\\mathrm{log}{\\left(\\frac{{x}^{3}\\left(x+5\\right)}{\\left(2x+3\\right)}\\right)}^{4}[\/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-gefccgbb-X_y9kQr7NRU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/X_y9kQr7NRU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gefccgbb-X_y9kQr7NRU\" 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=12850419&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gefccgbb-X_y9kQr7NRU&amp;vembed=0&amp;video_id=X_y9kQr7NRU&amp;video_target=tpm-plugin-gefccgbb-X_y9kQr7NRU\" 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+-+Expand+Logarithmic+Expressions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Expand Logarithmic Expressions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Using the Change-of-Base Formula for 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\">Change-of-Base Formula: For any positive real numbers [latex]M[\/latex], [latex]b[\/latex], and [latex]n[\/latex], where [latex]n \\neq 1[\/latex] and [latex]b \\neq 1[\/latex]: [latex]\\log_b M = \\frac{\\log_n M}{\\log_n b}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Common Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Using natural logarithms: [latex]\\log_b M = \\frac{\\ln M}{\\ln b}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using common logarithms: [latex]\\log_b M = \\frac{\\log M}{\\log b}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Purpose:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Evaluate logarithms with bases not available on standard calculators<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Convert between different logarithmic bases<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Change [latex]{\\mathrm{log}}_{0.5}8[\/latex] to a quotient of natural logarithms.[reveal-answer q=\"7928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"7928\"][latex]\\frac{\\mathrm{ln}8}{\\mathrm{ln}0.5}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]{\\mathrm{log}}_{5}\\left(100\\right)[\/latex] using the change-of-base formula.[reveal-answer q=\"732930\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"732930\"][latex]\\frac{\\mathrm{ln}100}{\\mathrm{ln}5}\\approx \\frac{4.6051}{1.6094}=2.861[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Expand logarithmic expressions.<\/li>\n<li>Condense logarithmic expressions.<\/li>\n<li>Use the change-of-base formula for logarithms.<\/li>\n<\/ul>\n<\/section>\n<h2>Basic Properties of 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\">Logarithmic-Exponential 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 \\Leftrightarrow b^y = x[\/latex], where [latex]b > 0, b \\neq 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Zero Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(1) = 0[\/latex] for any base [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Equivalent to [latex]b^0 = 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identity Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(b) = 1[\/latex] for any base [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Equivalent to [latex]b^1 = b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]b^{\\log_b(x)} = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(b^x) = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain Restrictions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]b > 0, b \\neq 1[\/latex] (base restriction)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x > 0[\/latex] (argument restriction)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Using the Product Rule for 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\">Product Rule Statement:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex], where [latex]b > 0, b \\neq 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relation to Exponent Rules:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Analogous to [latex]x^a \\cdot x^b = x^{a+b}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiple Factors:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(wxyz) = \\log_b(w) + \\log_b(x) + \\log_b(y) + \\log_b(z)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex]{\\mathrm{log}}_{b}\\left(8k\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q829261\">Show Solution<\/button><\/p>\n<div id=\"q829261\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\mathrm{log}}_{b}8+{\\mathrm{log}}_{b}k[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Using the Quotient Rule for 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\">Quotient Rule Statement:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(\\frac{M}{N}) = \\log_b(M) - \\log_b(N)[\/latex], where [latex]b > 0, b \\neq 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relation to Exponent Rules:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Analogous to [latex]\\frac{x^a}{x^b} = x^{a-b}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Combination with Product Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Often used in conjunction with the product rule for full expansion<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex]{\\mathrm{log}}_{3}\\left(\\frac{7{x}^{2}+21x}{7x\\left(x - 1\\right)\\left(x - 2\\right)}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q442008\">Show Solution<\/button><\/p>\n<div id=\"q442008\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\mathrm{log}}_{3}\\left(x+3\\right)-{\\mathrm{log}}_{3}\\left(x - 1\\right)-{\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Using the Power Rule for 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\">Power Rule Statement:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(M^n) = n \\log_b(M)[\/latex], where [latex]b > 0, b \\neq 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivation from Product Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(x^2) = \\log_b(x \\cdot x) = \\log_b(x) + \\log_b(x) = 2\\log_b(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Application to Roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(\\sqrt{x}) = \\log_b(x^{\\frac{1}{2}}) = \\frac{1}{2}\\log_b(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\mathrm{ln}{x}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q383972\">Show Solution<\/button><\/p>\n<div id=\"q383972\" class=\"hidden-answer\" style=\"display: none\">[latex]2\\mathrm{ln}x[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\mathrm{ln}\\left(\\frac{1}{{x}^{2}}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q947582\">Show Solution<\/button><\/p>\n<div id=\"q947582\" class=\"hidden-answer\" style=\"display: none\">[latex]-2\\mathrm{ln}\\left(x\\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-fbebcceb-SxF44olWTyk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SxF44olWTyk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbebcceb-SxF44olWTyk\" 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=12850417&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbebcceb-SxF44olWTyk&amp;vembed=0&amp;video_id=SxF44olWTyk&amp;video_target=tpm-plugin-fbebcceb-SxF44olWTyk\" 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\/The+Properties+of+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Properties of Logarithms\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-afefbgch-OiVILGg227E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/OiVILGg227E?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-afefbgch-OiVILGg227E\" 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=12850418&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-afefbgch-OiVILGg227E&amp;vembed=0&amp;video_id=OiVILGg227E&amp;video_target=tpm-plugin-afefbgch-OiVILGg227E\" 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+-+Evaluate+a+Natural+Logarithmic+Expression+Using+the+Properties+of+Logarithms_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Evaluate a Natural Logarithmic Expression Using the Properties of Logarithms\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Expanding and Condensing 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\">Expanding Logarithms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Apply rules to break down complex logarithmic expressions<\/li>\n<li class=\"whitespace-normal break-words\">Can only expand products, quotients, powers, and roots inside the logarithm<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Condensing Logarithms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combine multiple logarithms with the same base into a single logarithm<\/li>\n<li class=\"whitespace-normal break-words\">Apply rules in reverse order: Power rule first, then product\/quotient rules<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Important Considerations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Logarithms must have the same base to be combined<\/li>\n<li class=\"whitespace-normal break-words\">Addition or subtraction inside the logarithm cannot be expanded<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>Tip for success<\/strong><\/p>\n<ul>\n<li>When expanding and condensing logarithms, keep in mind that there are often more than one or two good ways to reach a good conclusion. The rules for manipulating exponents and logarithms can be combined creatively. You should try a few different ideas for using the rules on complicated expressions to get practice for finding the most efficient path to take in different situations.<\/li>\n<li>When working with more complicated examples, write your work down step by step to avoid making incorrect assumptions. It&#8217;s okay to try different rules as you practice creativity as long as you use each rule correctly.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex]\\mathrm{log}\\left(\\frac{{x}^{2}{y}^{3}}{{z}^{4}}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q722800\">Show Solution<\/button><\/p>\n<div id=\"q722800\" class=\"hidden-answer\" style=\"display: none\">[latex]2\\mathrm{log}x+3\\mathrm{log}y - 4\\mathrm{log}z[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex]\\mathrm{ln}\\left(\\sqrt[3]{{x}^{2}}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q2296\">Show Solution<\/button><\/p>\n<div id=\"q2296\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{2}{3}\\mathrm{ln}x[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex]\\mathrm{ln}\\left(\\frac{\\sqrt{\\left(x - 1\\right){\\left(2x+1\\right)}^{2}}}{\\left({x}^{2}-9\\right)}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q767425\">Show Solution<\/button><\/p>\n<div id=\"q767425\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{2}\\mathrm{ln}\\left(x - 1\\right)+\\mathrm{ln}\\left(2x+1\\right)-\\mathrm{ln}\\left(x+3\\right)-\\mathrm{ln}\\left(x - 3\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use the power rule for logs to rewrite [latex]2{\\mathrm{log}}_{3}4[\/latex] as a single logarithm with a leading coefficient of 1.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q720709\">Show Solution<\/button><\/p>\n<div id=\"q720709\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\mathrm{log}}_{3}16[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Condense [latex]\\mathrm{log}3-\\mathrm{log}4+\\mathrm{log}5-\\mathrm{log}6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q52020\">Show Solution<\/button><\/p>\n<div id=\"q52020\" class=\"hidden-answer\" style=\"display: none\">[latex]\\mathrm{log}\\left(\\frac{3\\cdot 5}{4\\cdot 6}\\right)[\/latex]; can also be written [latex]\\mathrm{log}\\left(\\frac{5}{8}\\right)[\/latex] by simplifying the fraction to lowest terms.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\mathrm{log}\\left(5\\right)+0.5\\mathrm{log}\\left(x\\right)-\\mathrm{log}\\left(7x - 1\\right)+3\\mathrm{log}\\left(x - 1\\right)[\/latex] as a single logarithm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q812624\">Show Solution<\/button><\/p>\n<div id=\"q812624\" class=\"hidden-answer\" style=\"display: none\">[latex]\\mathrm{log}\\left(\\frac{5{\\left(x - 1\\right)}^{3}\\sqrt{x}}{\\left(7x - 1\\right)}\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Condense [latex]4\\left(3\\mathrm{log}\\left(x\\right)+\\mathrm{log}\\left(x+5\\right)-\\mathrm{log}\\left(2x+3\\right)\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q622494\">Show Solution<\/button><\/p>\n<div id=\"q622494\" class=\"hidden-answer\" style=\"display: none\">[latex]\\mathrm{log}\\frac{{x}^{12}{\\left(x+5\\right)}^{4}}{{\\left(2x+3\\right)}^{4}}[\/latex]; this answer could also be written as [latex]\\mathrm{log}{\\left(\\frac{{x}^{3}\\left(x+5\\right)}{\\left(2x+3\\right)}\\right)}^{4}[\/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-gefccgbb-X_y9kQr7NRU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/X_y9kQr7NRU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gefccgbb-X_y9kQr7NRU\" 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=12850419&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gefccgbb-X_y9kQr7NRU&amp;vembed=0&amp;video_id=X_y9kQr7NRU&amp;video_target=tpm-plugin-gefccgbb-X_y9kQr7NRU\" 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+-+Expand+Logarithmic+Expressions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Expand Logarithmic Expressions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Using the Change-of-Base Formula for 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\">Change-of-Base Formula: For any positive real numbers [latex]M[\/latex], [latex]b[\/latex], and [latex]n[\/latex], where [latex]n \\neq 1[\/latex] and [latex]b \\neq 1[\/latex]: [latex]\\log_b M = \\frac{\\log_n M}{\\log_n b}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Common Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Using natural logarithms: [latex]\\log_b M = \\frac{\\ln M}{\\ln b}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Using common logarithms: [latex]\\log_b M = \\frac{\\log M}{\\log b}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Purpose:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Evaluate logarithms with bases not available on standard calculators<\/li>\n<li class=\"whitespace-normal break-words\">Convert between different logarithmic bases<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Change [latex]{\\mathrm{log}}_{0.5}8[\/latex] to a quotient of natural logarithms.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q7928\">Show Solution<\/button><\/p>\n<div id=\"q7928\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{\\mathrm{ln}8}{\\mathrm{ln}0.5}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate [latex]{\\mathrm{log}}_{5}\\left(100\\right)[\/latex] using the change-of-base formula.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q732930\">Show Solution<\/button><\/p>\n<div id=\"q732930\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{\\mathrm{ln}100}{\\mathrm{ln}5}\\approx \\frac{4.6051}{1.6094}=2.861[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"The Properties of Logarithms\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/SxF44olWTyk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Evaluate a Natural Logarithmic Expression Using the Properties of Logarithms\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/OiVILGg227E\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Expand Logarithmic Expressions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/X_y9kQr7NRU\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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