{"id":112,"date":"2025-02-13T22:43:54","date_gmt":"2025-02-13T22:43:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-equations\/"},"modified":"2026-03-18T18:17:36","modified_gmt":"2026-03-18T18:17:36","slug":"exponential-and-logarithmic-equations","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-equations\/","title":{"raw":"Exponential and Logarithmic Equations: Learn It 1","rendered":"Exponential and Logarithmic Equations: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><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\nThe first technique we will introduce for solving exponential equations involves two functions with like bases. The one-to-one property of exponential functions tells us that, for any real numbers [latex]b[\/latex], [latex]S[\/latex], and [latex]T[\/latex], where [latex]b&gt;0,\\text{ }b\\ne 1[\/latex], [latex]{b}^{S}={b}^{T}[\/latex] if and only if [latex]S\u00a0= T[\/latex].\r\n\r\nIn other words, when an <strong>exponential equation<\/strong> has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">For example, consider the equation [latex]{3}^{4x - 7}=\\frac{{3}^{2x}}{3}[\/latex]. To solve for [latex]x[\/latex], we use the division property of exponents to rewrite the right side so that both sides have the common base [latex]3[\/latex]. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for [latex]x[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{3}^{4x - 7}\\hfill &amp; =\\frac{{3}^{2x}}{3}\\hfill &amp; \\hfill \\\\ {3}^{4x - 7}\\hfill &amp; =\\frac{{3}^{2x}}{{3}^{1}}\\hfill &amp; {\\text{Rewrite 3 as 3}}^{1}.\\hfill \\\\ {3}^{4x - 7}\\hfill &amp; ={3}^{2x - 1}\\hfill &amp; \\text{Use the division property of exponents}\\text{.}\\hfill \\\\ 4x - 7\\hfill &amp; =2x - 1\\text{ }\\hfill &amp; \\text{Apply the one-to-one property of exponents}\\text{.}\\hfill \\\\ 2x\\hfill &amp; =6\\hfill &amp; \\text{Subtract 2}x\\text{ and add 7 to both sides}\\text{.}\\hfill \\\\ x\\hfill &amp; =3\\hfill &amp; \\text{Divide by 2}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>using the one-to-one property of exponential functions to solve exponential equations<\/h3>\r\nFor any algebraic expressions [latex]S[\/latex]\u00a0and [latex]T[\/latex], and any positive real number [latex]b\\ne 1[\/latex],\r\n<p style=\"text-align: center;\">[latex]{b}^{S}={b}^{T}\\text{ if and only if }S=T[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given an exponential equation Of the form [latex]{b}^{S}={b}^{T}[\/latex], where [latex]S[\/latex]\u00a0and\u00a0[latex]T[\/latex]\u00a0are algebraic expressions with an unknown, solve for the unknown<\/strong>\r\n<ol>\r\n \t<li>Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form [latex]{b}^{S}={b}^{T}[\/latex].<\/li>\r\n \t<li>Use the one-to-one property to set the exponents equal to each other.<\/li>\r\n \t<li>Solve the resulting equation, [latex]S\u00a0= T[\/latex], for the unknown.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{2}^{x - 1}={2}^{2x - 4}[\/latex].[reveal-answer q=\"766535\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"766535\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} {2}^{x - 1}={2}^{2x - 4}\\hfill &amp; \\text{The common base is }2.\\hfill \\\\ \\text{ }x - 1=2x - 4\\hfill &amp; \\text{By the one-to-one property the exponents must be equal}.\\hfill \\\\ \\text{ }x=3\\hfill &amp; \\text{Solve for }x.\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321519[\/ohm_question]<\/section>\r\n<h3>Rewriting Equations So All Powers Have the Same Base<\/h3>\r\nSometimes the <strong>common base<\/strong> for an exponential equation is not explicitly shown. In these cases we simply rewrite the terms in the equation as powers with a common base and solve using the one-to-one property.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{2}^{5x}=\\sqrt{2}[\/latex].[reveal-answer q=\"256816\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"256816\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{2}^{5x}={2}^{\\frac{1}{2}}\\hfill &amp; \\text{Write the square root of 2 as a power of }2.\\hfill \\\\ 5x=\\frac{1}{2}\\hfill &amp; \\text{Use the one-to-one property}.\\hfill \\\\ x=\\frac{1}{10}\\hfill &amp; \\text{Solve for }x.\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<div id=\"fs-id1165137730366\" class=\"solution\"><section id=\"fs-id1165137748966\"><section id=\"fs-id1165137667260\"><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\n[ohm_question hide_question_numbers=1]321520[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\n[ohm_question hide_question_numbers=1]321521[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section><\/section><\/section><\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">For example, consider the equation [latex]256={4}^{x - 5}[\/latex]. We can rewrite both sides of this equation as a power of [latex]2[\/latex]. Then we apply the rules of exponents along with the one-to-one property to solve for [latex]x[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}256={4}^{x - 5}\\hfill &amp; \\hfill \\\\ {2}^{8}={\\left({2}^{2}\\right)}^{x - 5}\\hfill &amp; \\text{Rewrite each side as a power with base 2}.\\hfill \\\\ {2}^{8}={2}^{2x - 10}\\hfill &amp; \\text{To take a power of a power, multiply the exponents}.\\hfill \\\\ 8=2x - 10\\hfill &amp; \\text{Apply the one-to-one property of exponents}.\\hfill \\\\ 18=2x\\hfill &amp; \\text{Add 10 to both sides}.\\hfill \\\\ x=9\\hfill &amp; \\text{Divide by 2}.\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given an exponential equation with unlike bases, use the one-to-one property to solve it<\/strong>\r\n<ol>\r\n \t<li>Rewrite each side in the equation as a power with a common base.<\/li>\r\n \t<li>Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form [latex]{b}^{S}={b}^{T}[\/latex].<\/li>\r\n \t<li>Use the one-to-one property to set the exponents equal to each other.<\/li>\r\n \t<li>Solve the resulting equation, [latex]S\u00a0= T[\/latex], for the unknown.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{8}^{x+2}={16}^{x+1}[\/latex].[reveal-answer q=\"214040\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"214040\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllll}\\text{ }{8}^{x+2}={16}^{x+1}\\hfill &amp; \\hfill \\\\ {\\left({2}^{3}\\right)}^{x+2}={\\left({2}^{4}\\right)}^{x+1}\\hfill &amp; \\text{Write }8\\text{ and }16\\text{ as powers of }2.\\hfill \\\\ \\text{ }{2}^{3x+6}={2}^{4x+4}\\hfill &amp; \\text{To take a power of a power, multiply the exponents}.\\hfill \\\\ \\text{ }3x+6=4x+4\\hfill &amp; \\text{Use the one-to-one property to set the exponents equal to each other}.\\hfill \\\\ \\text{ }x=2\\hfill &amp; \\text{Solve for }x.\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321522[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321523[\/ohm_question]<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?<\/strong>\r\n\r\n<hr \/>\r\n\r\nNo. Recall that the range of an exponential function is always positive. While solving the equation we may obtain an expression that is undefined.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{3}^{x+1}=-2[\/latex].[reveal-answer q=\"897533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"897533\"]This equation has no solution. There is no real value of [latex]x[\/latex]\u00a0that will make the equation a true statement because any power of a positive number is positive.\r\n[latex]\\\\[\/latex]\r\n<strong>Analysis of the Solution<\/strong>\r\n[latex]\\\\[\/latex]\r\nThe figure below\u00a0shows that the two graphs do not cross so the left side of the equation is never equal to the right side of the equation. Thus the equation has no solution.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03172009\/CNX_Precalc_Figure_04_06_0022.jpg\" alt=\"Graph of 3^(x+1)=-2 and y=-2. The graph notes that they do not cross.\" width=\"487\" height=\"438\" \/>[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321525[\/ohm_question]<\/section><\/div>\r\n<div id=\"fs-id1165137730366\" class=\"solution\"><section id=\"fs-id1165137748966\"><section id=\"fs-id1165137667260\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321526[\/ohm_question]<\/section><\/section><\/section><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<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<p>The first technique we will introduce for solving exponential equations involves two functions with like bases. The one-to-one property of exponential functions tells us that, for any real numbers [latex]b[\/latex], [latex]S[\/latex], and [latex]T[\/latex], where [latex]b>0,\\text{ }b\\ne 1[\/latex], [latex]{b}^{S}={b}^{T}[\/latex] if and only if [latex]S\u00a0= T[\/latex].<\/p>\n<p>In other words, when an <strong>exponential equation<\/strong> has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">For example, consider the equation [latex]{3}^{4x - 7}=\\frac{{3}^{2x}}{3}[\/latex]. To solve for [latex]x[\/latex], we use the division property of exponents to rewrite the right side so that both sides have the common base [latex]3[\/latex]. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for [latex]x[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{3}^{4x - 7}\\hfill & =\\frac{{3}^{2x}}{3}\\hfill & \\hfill \\\\ {3}^{4x - 7}\\hfill & =\\frac{{3}^{2x}}{{3}^{1}}\\hfill & {\\text{Rewrite 3 as 3}}^{1}.\\hfill \\\\ {3}^{4x - 7}\\hfill & ={3}^{2x - 1}\\hfill & \\text{Use the division property of exponents}\\text{.}\\hfill \\\\ 4x - 7\\hfill & =2x - 1\\text{ }\\hfill & \\text{Apply the one-to-one property of exponents}\\text{.}\\hfill \\\\ 2x\\hfill & =6\\hfill & \\text{Subtract 2}x\\text{ and add 7 to both sides}\\text{.}\\hfill \\\\ x\\hfill & =3\\hfill & \\text{Divide by 2}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>using the one-to-one property of exponential functions to solve exponential equations<\/h3>\n<p>For any algebraic expressions [latex]S[\/latex]\u00a0and [latex]T[\/latex], and any positive real number [latex]b\\ne 1[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]{b}^{S}={b}^{T}\\text{ if and only if }S=T[\/latex]<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given an exponential equation Of the form [latex]{b}^{S}={b}^{T}[\/latex], where [latex]S[\/latex]\u00a0and\u00a0[latex]T[\/latex]\u00a0are algebraic expressions with an unknown, solve for the unknown<\/strong><\/p>\n<ol>\n<li>Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form [latex]{b}^{S}={b}^{T}[\/latex].<\/li>\n<li>Use the one-to-one property to set the exponents equal to each other.<\/li>\n<li>Solve the resulting equation, [latex]S\u00a0= T[\/latex], for the unknown.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{2}^{x - 1}={2}^{2x - 4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q766535\">Show Solution<\/button><\/p>\n<div id=\"q766535\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} {2}^{x - 1}={2}^{2x - 4}\\hfill & \\text{The common base is }2.\\hfill \\\\ \\text{ }x - 1=2x - 4\\hfill & \\text{By the one-to-one property the exponents must be equal}.\\hfill \\\\ \\text{ }x=3\\hfill & \\text{Solve for }x.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321519\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321519&theme=lumen&iframe_resize_id=ohm321519&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Rewriting Equations So All Powers Have the Same Base<\/h3>\n<p>Sometimes the <strong>common base<\/strong> for an exponential equation is not explicitly shown. In these cases we simply rewrite the terms in the equation as powers with a common base and solve using the one-to-one property.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{2}^{5x}=\\sqrt{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q256816\">Show Solution<\/button><\/p>\n<div id=\"q256816\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{2}^{5x}={2}^{\\frac{1}{2}}\\hfill & \\text{Write the square root of 2 as a power of }2.\\hfill \\\\ 5x=\\frac{1}{2}\\hfill & \\text{Use the one-to-one property}.\\hfill \\\\ x=\\frac{1}{10}\\hfill & \\text{Solve for }x.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137730366\" class=\"solution\">\n<section id=\"fs-id1165137748966\">\n<section id=\"fs-id1165137667260\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p><iframe loading=\"lazy\" id=\"ohm321520\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321520&theme=lumen&iframe_resize_id=ohm321520&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p><iframe loading=\"lazy\" id=\"ohm321521\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321521&theme=lumen&iframe_resize_id=ohm321521&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/section>\n<\/section>\n<\/section>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">For example, consider the equation [latex]256={4}^{x - 5}[\/latex]. We can rewrite both sides of this equation as a power of [latex]2[\/latex]. Then we apply the rules of exponents along with the one-to-one property to solve for [latex]x[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}256={4}^{x - 5}\\hfill & \\hfill \\\\ {2}^{8}={\\left({2}^{2}\\right)}^{x - 5}\\hfill & \\text{Rewrite each side as a power with base 2}.\\hfill \\\\ {2}^{8}={2}^{2x - 10}\\hfill & \\text{To take a power of a power, multiply the exponents}.\\hfill \\\\ 8=2x - 10\\hfill & \\text{Apply the one-to-one property of exponents}.\\hfill \\\\ 18=2x\\hfill & \\text{Add 10 to both sides}.\\hfill \\\\ x=9\\hfill & \\text{Divide by 2}.\\hfill \\end{array}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given an exponential equation with unlike bases, use the one-to-one property to solve it<\/strong><\/p>\n<ol>\n<li>Rewrite each side in the equation as a power with a common base.<\/li>\n<li>Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form [latex]{b}^{S}={b}^{T}[\/latex].<\/li>\n<li>Use the one-to-one property to set the exponents equal to each other.<\/li>\n<li>Solve the resulting equation, [latex]S\u00a0= T[\/latex], for the unknown.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{8}^{x+2}={16}^{x+1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214040\">Show Solution<\/button><\/p>\n<div id=\"q214040\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllll}\\text{ }{8}^{x+2}={16}^{x+1}\\hfill & \\hfill \\\\ {\\left({2}^{3}\\right)}^{x+2}={\\left({2}^{4}\\right)}^{x+1}\\hfill & \\text{Write }8\\text{ and }16\\text{ as powers of }2.\\hfill \\\\ \\text{ }{2}^{3x+6}={2}^{4x+4}\\hfill & \\text{To take a power of a power, multiply the exponents}.\\hfill \\\\ \\text{ }3x+6=4x+4\\hfill & \\text{Use the one-to-one property to set the exponents equal to each other}.\\hfill \\\\ \\text{ }x=2\\hfill & \\text{Solve for }x.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321522\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321522&theme=lumen&iframe_resize_id=ohm321522&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321523\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321523&theme=lumen&iframe_resize_id=ohm321523&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?<\/strong><\/p>\n<hr \/>\n<p>No. Recall that the range of an exponential function is always positive. While solving the equation we may obtain an expression that is undefined.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]{3}^{x+1}=-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q897533\">Show Solution<\/button><\/p>\n<div id=\"q897533\" class=\"hidden-answer\" style=\"display: none\">This equation has no solution. There is no real value of [latex]x[\/latex]\u00a0that will make the equation a true statement because any power of a positive number is positive.<br \/>\n[latex]\\\\[\/latex]<br \/>\n<strong>Analysis of the Solution<\/strong><br \/>\n[latex]\\\\[\/latex]<br \/>\nThe figure below\u00a0shows that the two graphs do not cross so the left side of the equation is never equal to the right side of the equation. Thus the equation has no solution.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03172009\/CNX_Precalc_Figure_04_06_0022.jpg\" alt=\"Graph of 3^(x+1)=-2 and y=-2. The graph notes that they do not cross.\" width=\"487\" height=\"438\" \/><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321525\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321525&theme=lumen&iframe_resize_id=ohm321525&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<div class=\"solution\">\n<section>\n<section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321526\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321526&theme=lumen&iframe_resize_id=ohm321526&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<\/section>\n<\/div>\n","protected":false},"author":6,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":510,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/112"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/112\/revisions"}],"predecessor-version":[{"id":5912,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/112\/revisions\/5912"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/510"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/112\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=112"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=112"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=112"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}