{"id":1443,"date":"2025-07-25T01:28:18","date_gmt":"2025-07-25T01:28:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1443"},"modified":"2026-03-24T07:35:01","modified_gmt":"2026-03-24T07:35:01","slug":"exponential-and-logarithmic-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-equations-fresh-take\/","title":{"raw":"Exponential and Logarithmic Equations: Fresh Take","rendered":"Exponential and Logarithmic Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use like bases to solve exponential equations.<\/li>\r\n \t<li>Use logarithms to solve exponential equations.<\/li>\r\n \t<li>Solve logarithmic equations<\/li>\r\n \t<li>Solve applied problems involving exponential and logarithmic equations.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Exponential Equations<\/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\">One-to-One Property of Exponential Functions: For any real numbers [latex]b[\/latex], [latex]S[\/latex], and [latex]T[\/latex], where [latex]b &gt; 0, b \\neq 1[\/latex]: [latex]b^S = b^T[\/latex] if and only if [latex]S = T[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Techniques:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Equations with like bases: Set the exponents equal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Equations with unlike bases: Rewrite with a common base<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Limitations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Not all exponential equations have solutions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The range of an exponential function is always positive<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Approach<\/strong><\/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\">Identify the base(s) in the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If bases are different, rewrite terms to have a common base<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply rules of exponents to simplify if necessary<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the one-to-one property to set exponents equal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation for the unknown<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check for extraneous solutions or no solution scenarios<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{5}^{2x}={5}^{3x+2}[\/latex].[reveal-answer q=\"902679\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"902679\"][latex]x=\u20132[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{5}^{2x}={25}^{3x+2}[\/latex].[reveal-answer q=\"660468\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"660468\"][latex]x=\u20131[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{5}^{x}=\\sqrt{5}[\/latex].[reveal-answer q=\"743764\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"743764\"][latex]x=\\frac{1}{2}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{2}^{x}=-100[\/latex].[reveal-answer q=\"161944\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161944\"]The equation has no solution.[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script><\/h2>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-heebghde-XBpg0PN6ZG4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/XBpg0PN6ZG4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-heebghde-XBpg0PN6ZG4\" 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=12850420&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-heebghde-XBpg0PN6ZG4&amp;vembed=0&amp;video_id=XBpg0PN6ZG4&amp;video_target=tpm-plugin-heebghde-XBpg0PN6ZG4\" 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+-+Solve+a+Basic+Exponential+Equation+Using+the+Definition+of+a+Logarithm_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Solve a Basic Exponential Equation Using the Definition of a Logarithm\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Using Logarithms to Solve Exponential Equations<\/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\">Property of Logarithmic Equality: For [latex]M &gt; 0, N &gt; 0, b &gt; 0, b \\neq 1[\/latex]: If [latex]\\log_b(M) = \\log_b(N)[\/latex], then [latex]M = N[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Techniques:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Apply logarithms to both sides of the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use common log (base [latex]10[\/latex]) or natural log (base [latex]e[\/latex]) as needed<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Utilize logarithm rules to simplify and solve<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Special Cases:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Equations with base [latex]e[\/latex]: Use natural logarithm<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Watch for extraneous solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Approach<\/strong><\/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\">Identify if a common base can be found (if not, proceed with logarithms)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Choose appropriate logarithm (common or natural)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the logarithm to both sides of the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use logarithm rules to simplify<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for the unknown<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check for extraneous solutions<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{2}^{x}={3}^{x+1}[\/latex].[reveal-answer q=\"311643\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"311643\"][latex]x=\\frac{\\mathrm{ln}3}{\\mathrm{ln}\\left(\\frac{2}{3}\\right)}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]3{e}^{0.5t}=11[\/latex].[reveal-answer q=\"585330\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"585330\"][latex]t=2\\mathrm{ln}\\left(\\frac{11}{3}\\right)[\/latex] or [latex]\\mathrm{ln}{\\left(\\frac{11}{3}\\right)}^{2}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]3+{e}^{2t}=7{e}^{2t}[\/latex].[reveal-answer q=\"326491\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"326491\"][latex]t=\\mathrm{ln}\\left(\\frac{1}{\\sqrt{2}}\\right)=-\\frac{1}{2}\\mathrm{ln}\\left(2\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{e}^{2x}={e}^{x}+2[\/latex].[reveal-answer q=\"728419\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"728419\"][latex]x=\\mathrm{ln}2[\/latex][\/hidden-answer]<\/section>\r\n<h2>Logarithmic Equations<\/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: Every logarithmic equation [latex]\\log_b(x) = y[\/latex] is equivalent to the exponential equation [latex]b^y = x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Techniques:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Use logarithm rules to simplify expressions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Convert logarithmic equations to exponential form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate the logarithmic term before converting to exponential form<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Important Consideration:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Check for extraneous solutions, as logarithms are only defined for positive arguments<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Approach<\/strong><\/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\">Simplify the logarithmic expression using log rules if necessary<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate the logarithmic term on one side of the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Convert the equation to exponential form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting exponential equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check the solution(s) in the original equation to avoid extraneous solutions<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]6+\\mathrm{ln}x=10[\/latex].[reveal-answer q=\"183568\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"183568\"][latex]x={e}^{4}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]2\\mathrm{ln}\\left(x+1\\right)=10[\/latex].[reveal-answer q=\"62905\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"62905\"][latex]x={e}^{5}-1[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Use a graphing calculator to estimate the approximate solution to the logarithmic equation [latex]{2}^{x}=1000[\/latex] to 2 decimal places.[reveal-answer q=\"889911\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"889911\"][latex]x\\approx 9.97[\/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-hbbcfaff-ZUREfaCqWzY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ZUREfaCqWzY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hbbcfaff-ZUREfaCqWzY\" 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=12850421&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hbbcfaff-ZUREfaCqWzY&amp;vembed=0&amp;video_id=ZUREfaCqWzY&amp;video_target=tpm-plugin-hbbcfaff-ZUREfaCqWzY\" 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\/Solving+Logarithmic+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Logarithmic Equations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Using the One-to-One Property of Logarithms to Solve Logarithmic Equations<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p class=\"whitespace-pre-wrap break-words\">The one-to-one property of logarithms states that for any real numbers [latex]S &gt; 0[\/latex], [latex]T &gt; 0[\/latex], and any positive real number [latex]b \u2260 1[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\log_b S = \\log_b T \\text{ if and only if } S = T[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">This property allows us to solve logarithmic equations by equating the arguments when the bases are the same.<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/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\">Simplify logarithmic expressions using log rules<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the one-to-one property to equate arguments<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation for the unknown<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check for extraneous solutions<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Approach<\/strong><\/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\">Combine like terms using logarithm rules to get the equation in the form [latex]\\log_b S = \\log_b T[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the one-to-one property to set [latex]S = T[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation for the unknown<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check the solution(s) in the original equation to avoid extraneous solutions<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]\\mathrm{ln}\\left({x}^{2}\\right)=\\mathrm{ln}1[\/latex].[reveal-answer q=\"807762\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"807762\"][latex]x=1[\/latex] or [latex]x=\u20131[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use like bases to solve exponential equations.<\/li>\n<li>Use logarithms to solve exponential equations.<\/li>\n<li>Solve logarithmic equations<\/li>\n<li>Solve applied problems involving exponential and logarithmic equations.<\/li>\n<\/ul>\n<\/section>\n<h2>Exponential Equations<\/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\">One-to-One Property of Exponential Functions: For any real numbers [latex]b[\/latex], [latex]S[\/latex], and [latex]T[\/latex], where [latex]b > 0, b \\neq 1[\/latex]: [latex]b^S = b^T[\/latex] if and only if [latex]S = T[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Techniques:\n<ul>\n<li class=\"whitespace-normal break-words\">Equations with like bases: Set the exponents equal<\/li>\n<li class=\"whitespace-normal break-words\">Equations with unlike bases: Rewrite with a common base<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Limitations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Not all exponential equations have solutions<\/li>\n<li class=\"whitespace-normal break-words\">The range of an exponential function is always positive<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Approach<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the base(s) in the equation<\/li>\n<li class=\"whitespace-normal break-words\">If bases are different, rewrite terms to have a common base<\/li>\n<li class=\"whitespace-normal break-words\">Apply rules of exponents to simplify if necessary<\/li>\n<li class=\"whitespace-normal break-words\">Use the one-to-one property to set exponents equal<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation for the unknown<\/li>\n<li class=\"whitespace-normal break-words\">Check for extraneous solutions or no solution scenarios<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{5}^{2x}={5}^{3x+2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q902679\">Show Solution<\/button><\/p>\n<div id=\"q902679\" class=\"hidden-answer\" style=\"display: none\">[latex]x=\u20132[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{5}^{2x}={25}^{3x+2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q660468\">Show Solution<\/button><\/p>\n<div id=\"q660468\" class=\"hidden-answer\" style=\"display: none\">[latex]x=\u20131[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{5}^{x}=\\sqrt{5}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q743764\">Show Solution<\/button><\/p>\n<div id=\"q743764\" class=\"hidden-answer\" style=\"display: none\">[latex]x=\\frac{1}{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{2}^{x}=-100[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q161944\">Show Solution<\/button><\/p>\n<div id=\"q161944\" class=\"hidden-answer\" style=\"display: none\">The equation has no solution.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/h2>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-heebghde-XBpg0PN6ZG4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/XBpg0PN6ZG4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-heebghde-XBpg0PN6ZG4\" 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=12850420&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-heebghde-XBpg0PN6ZG4&amp;vembed=0&amp;video_id=XBpg0PN6ZG4&amp;video_target=tpm-plugin-heebghde-XBpg0PN6ZG4\" 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+-+Solve+a+Basic+Exponential+Equation+Using+the+Definition+of+a+Logarithm_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Solve a Basic Exponential Equation Using the Definition of a Logarithm\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Using Logarithms to Solve Exponential Equations<\/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\">Property of Logarithmic Equality: For [latex]M > 0, N > 0, b > 0, b \\neq 1[\/latex]: If [latex]\\log_b(M) = \\log_b(N)[\/latex], then [latex]M = N[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Techniques:\n<ul>\n<li class=\"whitespace-normal break-words\">Apply logarithms to both sides of the equation<\/li>\n<li class=\"whitespace-normal break-words\">Use common log (base [latex]10[\/latex]) or natural log (base [latex]e[\/latex]) as needed<\/li>\n<li class=\"whitespace-normal break-words\">Utilize logarithm rules to simplify and solve<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Special Cases:\n<ul>\n<li class=\"whitespace-normal break-words\">Equations with base [latex]e[\/latex]: Use natural logarithm<\/li>\n<li class=\"whitespace-normal break-words\">Watch for extraneous solutions<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Approach<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify if a common base can be found (if not, proceed with logarithms)<\/li>\n<li class=\"whitespace-normal break-words\">Choose appropriate logarithm (common or natural)<\/li>\n<li class=\"whitespace-normal break-words\">Apply the logarithm to both sides of the equation<\/li>\n<li class=\"whitespace-normal break-words\">Use logarithm rules to simplify<\/li>\n<li class=\"whitespace-normal break-words\">Solve for the unknown<\/li>\n<li class=\"whitespace-normal break-words\">Check for extraneous solutions<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{2}^{x}={3}^{x+1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q311643\">Show Solution<\/button><\/p>\n<div id=\"q311643\" class=\"hidden-answer\" style=\"display: none\">[latex]x=\\frac{\\mathrm{ln}3}{\\mathrm{ln}\\left(\\frac{2}{3}\\right)}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]3{e}^{0.5t}=11[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q585330\">Show Solution<\/button><\/p>\n<div id=\"q585330\" class=\"hidden-answer\" style=\"display: none\">[latex]t=2\\mathrm{ln}\\left(\\frac{11}{3}\\right)[\/latex] or [latex]\\mathrm{ln}{\\left(\\frac{11}{3}\\right)}^{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]3+{e}^{2t}=7{e}^{2t}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q326491\">Show Solution<\/button><\/p>\n<div id=\"q326491\" class=\"hidden-answer\" style=\"display: none\">[latex]t=\\mathrm{ln}\\left(\\frac{1}{\\sqrt{2}}\\right)=-\\frac{1}{2}\\mathrm{ln}\\left(2\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{e}^{2x}={e}^{x}+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q728419\">Show Solution<\/button><\/p>\n<div id=\"q728419\" class=\"hidden-answer\" style=\"display: none\">[latex]x=\\mathrm{ln}2[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Logarithmic Equations<\/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: Every logarithmic equation [latex]\\log_b(x) = y[\/latex] is equivalent to the exponential equation [latex]b^y = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Techniques:\n<ul>\n<li class=\"whitespace-normal break-words\">Use logarithm rules to simplify expressions<\/li>\n<li class=\"whitespace-normal break-words\">Convert logarithmic equations to exponential form<\/li>\n<li class=\"whitespace-normal break-words\">Isolate the logarithmic term before converting to exponential form<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Important Consideration:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Check for extraneous solutions, as logarithms are only defined for positive arguments<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Approach<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplify the logarithmic expression using log rules if necessary<\/li>\n<li class=\"whitespace-normal break-words\">Isolate the logarithmic term on one side of the equation<\/li>\n<li class=\"whitespace-normal break-words\">Convert the equation to exponential form<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting exponential equation<\/li>\n<li class=\"whitespace-normal break-words\">Check the solution(s) in the original equation to avoid extraneous solutions<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]6+\\mathrm{ln}x=10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q183568\">Show Solution<\/button><\/p>\n<div id=\"q183568\" class=\"hidden-answer\" style=\"display: none\">[latex]x={e}^{4}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]2\\mathrm{ln}\\left(x+1\\right)=10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q62905\">Show Solution<\/button><\/p>\n<div id=\"q62905\" class=\"hidden-answer\" style=\"display: none\">[latex]x={e}^{5}-1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use a graphing calculator to estimate the approximate solution to the logarithmic equation [latex]{2}^{x}=1000[\/latex] to 2 decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q889911\">Show Solution<\/button><\/p>\n<div id=\"q889911\" class=\"hidden-answer\" style=\"display: none\">[latex]x\\approx 9.97[\/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-hbbcfaff-ZUREfaCqWzY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ZUREfaCqWzY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hbbcfaff-ZUREfaCqWzY\" 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=12850421&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hbbcfaff-ZUREfaCqWzY&amp;vembed=0&amp;video_id=ZUREfaCqWzY&amp;video_target=tpm-plugin-hbbcfaff-ZUREfaCqWzY\" 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\/Solving+Logarithmic+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Logarithmic Equations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Using the One-to-One Property of Logarithms to Solve Logarithmic Equations<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">The one-to-one property of logarithms states that for any real numbers [latex]S > 0[\/latex], [latex]T > 0[\/latex], and any positive real number [latex]b \u2260 1[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\log_b S = \\log_b T \\text{ if and only if } S = T[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">This property allows us to solve logarithmic equations by equating the arguments when the bases are the same.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplify logarithmic expressions using log rules<\/li>\n<li class=\"whitespace-normal break-words\">Apply the one-to-one property to equate arguments<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation for the unknown<\/li>\n<li class=\"whitespace-normal break-words\">Check for extraneous solutions<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Approach<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combine like terms using logarithm rules to get the equation in the form [latex]\\log_b S = \\log_b T[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply the one-to-one property to set [latex]S = T[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation for the unknown<\/li>\n<li class=\"whitespace-normal break-words\">Check the solution(s) in the original equation to avoid extraneous solutions<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]\\mathrm{ln}\\left({x}^{2}\\right)=\\mathrm{ln}1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q807762\">Show Solution<\/button><\/p>\n<div id=\"q807762\" class=\"hidden-answer\" style=\"display: none\">[latex]x=1[\/latex] or [latex]x=\u20131[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Solve a Basic Exponential Equation Using the Definition of a Logarithm\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/XBpg0PN6ZG4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Solving Logarithmic Equations\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/ZUREfaCqWzY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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