{"id":15944,"date":"2019-09-25T21:22:59","date_gmt":"2019-09-25T21:22:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/use-compound-interest-formulas\/"},"modified":"2024-05-02T16:55:45","modified_gmt":"2024-05-02T16:55:45","slug":"use-compound-interest-formulas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/use-compound-interest-formulas\/","title":{"raw":"Exponential Equations with Like Bases","rendered":"Exponential Equations with Like Bases"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify an exponential equation whose terms all have the same base<\/li>\r\n \t<li>Identify cases where equations can be rewritten so all terms have the same base<\/li>\r\n \t<li>Apply the one-to-one property of exponents to solve an exponential equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165134354674\">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 can\u00a0set the exponents equal to one another and solve for the unknown.<\/p>\r\n<p id=\"fs-id1165135192889\">For example, consider the equation [latex]{3}^{4x - 7}=\\frac{{3}^{2x}}{3}[\/latex]. To solve for <i>x<\/i>, we use the division property of exponents to rewrite the right side so that both sides have the common base,\u00a0[latex]3[\/latex]. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for <em>x<\/em>:<\/p>\r\n\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}{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]<\/div>\r\nIn our first example, we solve an exponential equation whose terms all have a common base.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]{2}^{x - 1}={2}^{2x - 4}[\/latex].\r\n[reveal-answer q=\"579160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"579160\"]\r\n\r\n[latex]\\begin{array}{c} {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]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn general, we can summarize solving exponential equations whose terms all have the same base in this way:\r\nFor any algebraic expressions <em>S<\/em>\u00a0and <em>T<\/em>, and any positive real number [latex]b\\ne 1[\/latex]\r\n<div id=\"fs-id1165137702126\" class=\"equation\">[latex]{b}^{S}={b}^{T}\\text{ if and only if }S=T[\/latex]<\/div>\r\n<div class=\"equation\">\r\n<ul>\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, <em>S\u00a0<\/em>= <em>T<\/em>, for the unknown.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137730366\" class=\"solution\"><section><section id=\"fs-id1165137667260\">\r\n<h2>Rewriting Equations So All Powers Have the Same Base<\/h2>\r\n<p id=\"fs-id1165137725147\">Sometimes\u00a0we can rewrite the terms in an equation as powers with a common base and solve using the one-to-one property. This takes a keen eye for recognizing common powers. \u00a0For example, you can rewrite 8 as [latex]2^3[\/latex] or 36 as [latex]6^2[\/latex] or [latex]\\frac{1}{4}[\/latex] as [latex]\\left(\\frac{1}{2}\\right)^{2}[\/latex]<\/p>\r\n<p id=\"fs-id1165137784867\">Consider the equation [latex]256={4}^{x - 5}[\/latex]. We can rewrite both sides of this equation as a power of\u00a0[latex]2[\/latex]. Then we apply the rules of exponents, along with the one-to-one property, to solve for <em>x<\/em>:<\/p>\r\n\r\n<div id=\"eip-687\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}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{Use the power to a power property of 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]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">In the next example, we show how to find a common base for two expressions whose bases are\u00a0[latex]8[\/latex] and\u00a0[latex]16[\/latex]. We can then solve the resulting equation using the one-to-one property of exponents.<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]{8}^{x+2}={16}^{x+1}[\/latex].\r\n[reveal-answer q=\"731579\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"731579\"][latex]\\begin{array}{c}\\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 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]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our next example, we are given an exponential equation that contains a square root. \u00a0Remember that you can write roots as rational exponents, so you may be able to find like bases when it is not completely obvious at first.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]{2}^{5x}=\\sqrt{2}[\/latex].\r\n[reveal-answer q=\"507738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507738\"]\r\n\r\n[latex]\\begin{array}{c}{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]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nBy changing [latex]\\sqrt{2}[\/latex] to [latex]{2}^{\\frac{1}{2}}[\/latex], we were able to solve the equation in the previous example. In general, here are some steps to consider when you are solving exponential equations. \u00a0A good first step is always to determine whether you can rewrite the terms with a common base.\r\n<ol id=\"fs-id1165137663646\">\r\n \t<li>Rewrite each side of 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, <em>S\u00a0<\/em>= <em>T<\/em>, for the unknown.<\/li>\r\n<\/ol>\r\nIn the following video, we show more examples of how to solve exponential equations by finding a common base.\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=aPyE9SKtczs[\/embed]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nDo all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process? Write your thoughts in the textbox below before you check our proposed answer.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"711116\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"711116\"]\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<em>.<\/em>[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example, we show you a case where there is no solution to an exponential equation. Remember how exponential functions are defined and ask yourself, \"does this make sense\", before diving into solving exponential equations.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]{3}^{x+1}=-2[\/latex].\r\n[reveal-answer q=\"152201\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"152201\"]\r\n\r\nThis equation has no solution. There is no real value of <em>x<\/em>\u00a0that will make the equation a true statement because any power of a positive number is positive.\r\n\r\nFor example [latex]3^2=9[\/latex] and [latex]2^4=16[\/latex]. Remember that we have defined exponential functions as having a base that is greater than 0.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>\u00a0Analysis of the Solution<\/h3>\r\n<\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"Example_04_06_04\" class=\"example\">\r\n<div id=\"fs-id1165137405247\" class=\"exercise\">\r\n<div id=\"fs-id1165137849213\" class=\"commentary\">\r\n<p id=\"fs-id1165137578263\">The figure below\u00a0shows the graphs of the two separate expressions in the equation [latex]{3}^{x+1}=-2[\/latex] as [latex]y={3}^{x+1}[\/latex] and [latex]y=-2[\/latex]. The two graphs do not cross showing us that\u00a0the left side is never equal to the right side. Thus the equation has no solution.<\/p>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201934\/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\" \/>\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nWe can use the one-to-one property of exponents to solve exponential equations whose bases are the same. \u00a0The terms in some exponential equations can be rewritten with the same base, allowing us to use the same principle. There are exponential equations that do not have solutions because we define exponential functions as having a positive base. When restrictions are placed on the inputs of a function, it is natural that there will be restrictions on the output as well.\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify an exponential equation whose terms all have the same base<\/li>\n<li>Identify cases where equations can be rewritten so all terms have the same base<\/li>\n<li>Apply the one-to-one property of exponents to solve an exponential equation<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165134354674\">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 can\u00a0set the exponents equal to one another and solve for the unknown.<\/p>\n<p id=\"fs-id1165135192889\">For example, consider the equation [latex]{3}^{4x - 7}=\\frac{{3}^{2x}}{3}[\/latex]. To solve for <i>x<\/i>, we use the division property of exponents to rewrite the right side so that both sides have the common base,\u00a0[latex]3[\/latex]. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for <em>x<\/em>:<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}{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]<\/div>\n<p>In our first example, we solve an exponential equation whose terms all have a common base.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]{2}^{x - 1}={2}^{2x - 4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q579160\">Show Solution<\/span><\/p>\n<div id=\"q579160\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{c} {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<\/div>\n<p>In general, we can summarize solving exponential equations whose terms all have the same base in this way:<br \/>\nFor any algebraic expressions <em>S<\/em>\u00a0and <em>T<\/em>, and any positive real number [latex]b\\ne 1[\/latex]<\/p>\n<div id=\"fs-id1165137702126\" class=\"equation\">[latex]{b}^{S}={b}^{T}\\text{ if and only if }S=T[\/latex]<\/div>\n<div class=\"equation\">\n<ul>\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, <em>S\u00a0<\/em>= <em>T<\/em>, for the unknown.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137730366\" class=\"solution\">\n<section>\n<section id=\"fs-id1165137667260\">\n<h2>Rewriting Equations So All Powers Have the Same Base<\/h2>\n<p id=\"fs-id1165137725147\">Sometimes\u00a0we can rewrite the terms in an equation as powers with a common base and solve using the one-to-one property. This takes a keen eye for recognizing common powers. \u00a0For example, you can rewrite 8 as [latex]2^3[\/latex] or 36 as [latex]6^2[\/latex] or [latex]\\frac{1}{4}[\/latex] as [latex]\\left(\\frac{1}{2}\\right)^{2}[\/latex]<\/p>\n<p id=\"fs-id1165137784867\">Consider the equation [latex]256={4}^{x - 5}[\/latex]. We can rewrite both sides of this equation as a power of\u00a0[latex]2[\/latex]. Then we apply the rules of exponents, along with the one-to-one property, to solve for <em>x<\/em>:<\/p>\n<div id=\"eip-687\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}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{Use the power to a power property of 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]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">In the next example, we show how to find a common base for two expressions whose bases are\u00a0[latex]8[\/latex] and\u00a0[latex]16[\/latex]. We can then solve the resulting equation using the one-to-one property of exponents.<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]{8}^{x+2}={16}^{x+1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q731579\">Show Solution<\/span><\/p>\n<div id=\"q731579\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{array}{c}\\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 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<\/div>\n<p>In our next example, we are given an exponential equation that contains a square root. \u00a0Remember that you can write roots as rational exponents, so you may be able to find like bases when it is not completely obvious at first.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]{2}^{5x}=\\sqrt{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q507738\">Show Solution<\/span><\/p>\n<div id=\"q507738\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{c}{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<\/div>\n<p>By changing [latex]\\sqrt{2}[\/latex] to [latex]{2}^{\\frac{1}{2}}[\/latex], we were able to solve the equation in the previous example. In general, here are some steps to consider when you are solving exponential equations. \u00a0A good first step is always to determine whether you can rewrite the terms with a common base.<\/p>\n<ol id=\"fs-id1165137663646\">\n<li>Rewrite each side of 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, <em>S\u00a0<\/em>= <em>T<\/em>, for the unknown.<\/li>\n<\/ol>\n<p>In the following video, we show more examples of how to solve exponential equations by finding a common base.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solving Exponential Equations - Part 1 of 2\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/aPyE9SKtczs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process? Write your thoughts in the textbox below before you check our proposed answer.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q711116\">Show Solution<\/span><\/p>\n<div id=\"q711116\" class=\"hidden-answer\" style=\"display: none\">\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<em>.<\/em><\/div>\n<\/div>\n<\/div>\n<p>In the next example, we show you a case where there is no solution to an exponential equation. Remember how exponential functions are defined and ask yourself, &#8220;does this make sense&#8221;, before diving into solving exponential equations.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]{3}^{x+1}=-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q152201\">Show Solution<\/span><\/p>\n<div id=\"q152201\" class=\"hidden-answer\" style=\"display: none\">\n<p>This equation has no solution. There is no real value of <em>x<\/em>\u00a0that will make the equation a true statement because any power of a positive number is positive.<\/p>\n<p>For example [latex]3^2=9[\/latex] and [latex]2^4=16[\/latex]. Remember that we have defined exponential functions as having a base that is greater than 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>\u00a0Analysis of the Solution<\/h3>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"Example_04_06_04\" class=\"example\">\n<div id=\"fs-id1165137405247\" class=\"exercise\">\n<div id=\"fs-id1165137849213\" class=\"commentary\">\n<p id=\"fs-id1165137578263\">The figure below\u00a0shows the graphs of the two separate expressions in the equation [latex]{3}^{x+1}=-2[\/latex] as [latex]y={3}^{x+1}[\/latex] and [latex]y=-2[\/latex]. The two graphs do not cross showing us that\u00a0the left side is never equal to the right side. Thus the equation has no solution.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201934\/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\" \/><\/p>\n<\/div>\n<h2>Summary<\/h2>\n<p>We can use the one-to-one property of exponents to solve exponential equations whose bases are the same. \u00a0The terms in some exponential equations can be rewritten with the same base, allowing us to use the same principle. There are exponential equations that do not have solutions because we define exponential functions as having a positive base. When restrictions are placed on the inputs of a function, it is natural that there will be restrictions on the output as well.<\/p>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15944\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"fb0e0e8d6ce24ba6acac79cd1cb1c5a7, edc92b286a7041a1908c16c420c5c1b9","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15944","chapter","type-chapter","status-publish","hentry"],"part":15967,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/15944","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/15944\/revisions"}],"predecessor-version":[{"id":19174,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/15944\/revisions\/19174"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/15967"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/15944\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=15944"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=15944"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=15944"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=15944"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}