{"id":2237,"date":"2022-05-20T17:04:51","date_gmt":"2022-05-20T17:04:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2237"},"modified":"2025-12-15T19:07:13","modified_gmt":"2025-12-15T19:07:13","slug":"5-4-algebraic-analysis-on-intersection-points","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/5-4-algebraic-analysis-on-intersection-points\/","title":{"raw":"5.4: Algebraic Analysis on Intersection Points","rendered":"5.4: Algebraic Analysis on Intersection Points"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Describe the meaning of solving exponential equations<\/li>\r\n \t<li>Use the property of equality to solve exponential equations<\/li>\r\n \t<li>Determine the intersection point of two exponential functions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Meaning of Solving Exponential Equations<\/h2>\r\n<h3>Intersection point of functions<\/h3>\r\nIn chapter 3, we learned that the meaning of solving an equation is to find the intersection point(s) between two functions.\u00a0The intersection point(s) between the graphs of any two functions [latex]f(x)[\/latex] and\u00a0[latex]g(x)[\/latex]\u00a0can be found algebraically by setting the two functions equal to each other:\r\n<p style=\"text-align: center;\">[latex]f(x)=g(x)[\/latex]<\/p>\r\nWhen the functions are equal, the value of [latex]x[\/latex]\u00a0is the same for both functions, as is the function value. In other words, [latex]f(x)=g(x)[\/latex]\u00a0means that the two functions have the same input\u00a0[latex]x[\/latex] as well as the same output (i.e\u00a0[latex]f(x)=g(x)[\/latex]).\u00a0For example, solving the equation [latex]32\\left(2^{x+3}\\right)=64[\/latex] means finding the intersection point between the two functions [latex]f(x)=32\\left(2^{x+3}\\right)[\/latex] and [latex]g(x)=64[\/latex] (figure 1).\r\n\r\n[caption id=\"attachment_2878\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2878 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/21041434\/desmos-graph-2022-06-20T221421.267-300x300.png\" alt=\"Intersection of two functions\" width=\"300\" height=\"300\" \/> FIgure 1. Intersection of [latex]y=f(x)[\/latex] and [latex]y=g(x)[\/latex][\/caption]\r\n<h4>Finding the [latex]x[\/latex]-value given a function value<\/h4>\r\nAnother interpretation of the equation [latex](32)2^{x+3}=64[\/latex] is finding the [latex]x[\/latex] value when the function value of [latex]f(x)=(32)2^{x+3}[\/latex] is 64 (figure 2).\r\n\r\n[caption id=\"attachment_2879\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2879 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/21042317\/desmos-graph-2022-06-20T222249.950-300x300.png\" alt=\"exponential equation meaning\" width=\"300\" height=\"300\" \/> Figure 2. Meaning of solving an equation[\/caption]\r\n\r\nGraphically, this means finding the function value as a given [latex]y[\/latex]-value on a graph, then moving vertically down to the [latex]x[\/latex]-axis to determine the corresponding [latex]x[\/latex]-value. In figure 2, to solve\u00a0[latex](32)2^{x+3}=64[\/latex], we graph the function\u00a0[latex]f(x)=(32)2^{x+3}[\/latex], look for 64 on the [latex]y[\/latex]-axis then determine which [latex]x[\/latex]-value has a function value at 64. In this case, [latex]x=-2[\/latex].\r\n<h3>Solving an equation in one variable<\/h3>\r\nAlgebraically, [latex](32)2^{x+3}=64[\/latex] is an equation in one variable. When solving an equation in one variable, we find the value of the variable that satisfies the equation (e.g., the [latex]x[\/latex]-value). There is no function value to report as the equation is in just one variable.\r\n\r\nFor example, when solving the equation, [latex]3^x=9[\/latex], we find the value of [latex]x[\/latex] that makes the equation true. The value of\u00a0[latex]x[\/latex] is 2 because [latex]3^2=9[\/latex]. We need algebraic methods to solve equations, including the properties of equality covered in chapter 3.\r\n\r\nIn summary, when we set functions equal to each other, we are finding the intersection point between two functions (e.g., [latex]f(x)=3^x[\/latex] and\u00a0[latex]g(x)=9[\/latex]). In this example, the intersection point between the graphs of the two functions [latex]f(x)[\/latex] and\u00a0[latex]g(x)[\/latex]\u00a0is (2, 9) because the [latex]y[\/latex]-value is the function value [latex]f(2) = 3^2 = 9[\/latex] and [latex]g(x)=9[\/latex]. However, when we are solving an equation algebraically, there is no function in sight so we can just report the value of the variable (e.g., [latex]x[\/latex]).\r\n<h2>Solving Exponential Equations Using the Property of Equality<\/h2>\r\nWe may use the <em><strong>property of equality<\/strong><\/em> for solving an exponential equation if the exponential expressions on each side of the equation have the same base.\r\n\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: left;\">Property of Equality<\/h3>\r\n<p style=\"text-align: center;\">For any real numbers [latex]a, \\;x[\/latex] and [latex]y[\/latex],<\/p>\r\n<p style=\"text-align: center;\">If [latex]a^x = a^y[\/latex], then [latex]x = y[\/latex].<\/p>\r\n\r\n<\/div>\r\nFor example, to solve the equation [latex]3^x = 9[\/latex], we can write [latex]9 = 3^2[\/latex] so that [latex]3^x = 3^2[\/latex]. The property of equality then tells us that [latex]x = 2[\/latex] because with the same base, the exponents must be equal. Although it may at first appear that the exponential expressions in an equation do not have the same base, they can often be transformed to expressions with the same base.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nSolve the equation [latex]32\\left(2^{x+3}\\right)=64[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}32\\left(2^{x+3}\\right)&amp;=64&amp;&amp;\\text{The base of the exponential is 2, so we try to write 32 as a power of 2}\\\\2^5\\times 2^{x+3}&amp;=2^6&amp;&amp;\\text{Product rule}\\\\2^{x+3+5}&amp;=2^6\\\\2^{x+8}&amp;=2^6&amp;&amp;\\text{Now each side has the same base}\\\\x+8&amp;=6&amp;&amp;\\text{The exponents must be equal}\\\\x&amp;=-2\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nSolve the equation [latex]32\\left(2^{x+3}\\right)=64[\/latex].\r\n<h4>Solution<\/h4>\r\nAnother, and more efficient way to solve the same equation as example 1, is to notice that each side of the equation is divisible by 32:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}32\\left(2^{x+3}\\right)&amp;=64&amp;&amp;\\text{Divide each side by 32}\\\\2^{x+3}&amp;=2&amp;&amp;\\text{Same base so exponents must be equal}\\\\x+3&amp;=1&amp;&amp;\\text{Subtract 3 from both sides}\\\\x&amp;=-2\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 1<\/h3>\r\nSolve the equation [latex]15\\left(3^{x+4}\\right)=45[\/latex].\r\n\r\n[reveal-answer q=\"hjm577\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm577\"]\r\n\r\n[latex]x=-3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nSolve the equation [latex]\\dfrac{1}{4}\\left(6^{2x-1}\\right)=9[\/latex]\r\n\r\n[reveal-answer q=\"hjm397\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm397\"]\r\n\r\n[latex]x=\\dfrac{3}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nSolve the equation [latex]27^{x+1}=81^{x-2}[\/latex].\r\n<h4>Solution<\/h4>\r\nThe bases of the exponential expressions are 27 and 81. Both of these can be written as a power of 3: [latex]27 = 3^3[\/latex] and\u00a0[latex]81 = 3^4[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}27^{x+1}&amp;=81^{x-2}\\\\{\\left(3^3\\right)}^{x+1}&amp;={\\left(3^4\\right)}^{x-2}&amp;&amp;\\text{Power to a power rule}\\\\3^{3(x+1)}&amp;=3^{4(x-2)}&amp;&amp;\\text{Same base so exponents must be equal}\\\\3(x+1)&amp;=4(x-2)&amp;&amp;\\text{Distribute}\\\\3x+3&amp;=4x-8&amp;&amp;\\text{Subtract }3x\\text{ from both sides}\\\\3&amp;=x-8&amp;&amp;\\text{Add 8 to both sides}\\\\11&amp;=x\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nSolve the equation\u00a0[latex]2^{-x+5}=16^{x+1}[\/latex].\r\n\r\n[reveal-answer q=\"Leo410\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"Leo410\"]\r\n\r\n[latex]x=\\dfrac{1}{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nDetermine the intersection point of the functions [latex]f(x)=5\\left(2^{x-3}\\right)[\/latex] and [latex]g(x)=10\\left(2^{-x+16}\\right)[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: left;\">The intersection point occurs at an [latex]x[\/latex]-value where [latex]f(x)=g(x)[\/latex].<\/p>\r\nSet\u00a0[latex]f(x)=g(x)[\/latex] and solve for [latex]x[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}5\\left(2^{x-3}\\right)&amp;=10\\left(2^{-x+16}\\right)\\\\2^{x-3}&amp;=2\\left(2^{-x+16}\\right)\\\\2^{x-3}&amp;=2^{-x+17}\\\\x-3&amp;=-x+17\\\\2x-3&amp;=17\\\\2x&amp;=20\\\\x&amp;=10\\end{aligned}[\/latex]<\/p>\r\nNow that we know the value of [latex]x[\/latex], we can find the corresponding value of [latex]y[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}y&amp;=f(10)\\\\&amp;=5\\left(2^{10-3}\\right)\\\\&amp;=5\\left(2^7\\right)\\\\&amp;=5(128)\\\\&amp;=640\\end{aligned}[\/latex]<\/p>\r\nConsequently, the intersection point is [latex](10, 640)[\/latex].\r\n\r\n&nbsp;\r\n\r\nNote: We could also have used [latex]g(x)[\/latex] to find the value of [latex]y[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}y&amp;=g(10)\\\\&amp;=10\\left(2^{-10+16}\\right)\\\\&amp;=10\\left(2^{6}\\right)\\\\&amp;=10\\cdot 64\\\\&amp;=640\\end{aligned}[\/latex]<\/p>\r\nThis is a good check for the answer.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nDetermine the intersection point of the functions [latex]f(x)=6\\left(3^x\\right)[\/latex] and [latex]g(x)=18\\left(3^{-x+1}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"hjm201\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm201\"]\r\n\r\n(1, 18)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Describe the meaning of solving exponential equations<\/li>\n<li>Use the property of equality to solve exponential equations<\/li>\n<li>Determine the intersection point of two exponential functions<\/li>\n<\/ul>\n<\/div>\n<h2>The Meaning of Solving Exponential Equations<\/h2>\n<h3>Intersection point of functions<\/h3>\n<p>In chapter 3, we learned that the meaning of solving an equation is to find the intersection point(s) between two functions.\u00a0The intersection point(s) between the graphs of any two functions [latex]f(x)[\/latex] and\u00a0[latex]g(x)[\/latex]\u00a0can be found algebraically by setting the two functions equal to each other:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=g(x)[\/latex]<\/p>\n<p>When the functions are equal, the value of [latex]x[\/latex]\u00a0is the same for both functions, as is the function value. In other words, [latex]f(x)=g(x)[\/latex]\u00a0means that the two functions have the same input\u00a0[latex]x[\/latex] as well as the same output (i.e\u00a0[latex]f(x)=g(x)[\/latex]).\u00a0For example, solving the equation [latex]32\\left(2^{x+3}\\right)=64[\/latex] means finding the intersection point between the two functions [latex]f(x)=32\\left(2^{x+3}\\right)[\/latex] and [latex]g(x)=64[\/latex] (figure 1).<\/p>\n<div id=\"attachment_2878\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2878\" class=\"wp-image-2878 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/21041434\/desmos-graph-2022-06-20T221421.267-300x300.png\" alt=\"Intersection of two functions\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2878\" class=\"wp-caption-text\">FIgure 1. Intersection of [latex]y=f(x)[\/latex] and [latex]y=g(x)[\/latex]<\/p>\n<\/div>\n<h4>Finding the [latex]x[\/latex]-value given a function value<\/h4>\n<p>Another interpretation of the equation [latex](32)2^{x+3}=64[\/latex] is finding the [latex]x[\/latex] value when the function value of [latex]f(x)=(32)2^{x+3}[\/latex] is 64 (figure 2).<\/p>\n<div id=\"attachment_2879\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2879\" class=\"wp-image-2879 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/21042317\/desmos-graph-2022-06-20T222249.950-300x300.png\" alt=\"exponential equation meaning\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2879\" class=\"wp-caption-text\">Figure 2. Meaning of solving an equation<\/p>\n<\/div>\n<p>Graphically, this means finding the function value as a given [latex]y[\/latex]-value on a graph, then moving vertically down to the [latex]x[\/latex]-axis to determine the corresponding [latex]x[\/latex]-value. In figure 2, to solve\u00a0[latex](32)2^{x+3}=64[\/latex], we graph the function\u00a0[latex]f(x)=(32)2^{x+3}[\/latex], look for 64 on the [latex]y[\/latex]-axis then determine which [latex]x[\/latex]-value has a function value at 64. In this case, [latex]x=-2[\/latex].<\/p>\n<h3>Solving an equation in one variable<\/h3>\n<p>Algebraically, [latex](32)2^{x+3}=64[\/latex] is an equation in one variable. When solving an equation in one variable, we find the value of the variable that satisfies the equation (e.g., the [latex]x[\/latex]-value). There is no function value to report as the equation is in just one variable.<\/p>\n<p>For example, when solving the equation, [latex]3^x=9[\/latex], we find the value of [latex]x[\/latex] that makes the equation true. The value of\u00a0[latex]x[\/latex] is 2 because [latex]3^2=9[\/latex]. We need algebraic methods to solve equations, including the properties of equality covered in chapter 3.<\/p>\n<p>In summary, when we set functions equal to each other, we are finding the intersection point between two functions (e.g., [latex]f(x)=3^x[\/latex] and\u00a0[latex]g(x)=9[\/latex]). In this example, the intersection point between the graphs of the two functions [latex]f(x)[\/latex] and\u00a0[latex]g(x)[\/latex]\u00a0is (2, 9) because the [latex]y[\/latex]-value is the function value [latex]f(2) = 3^2 = 9[\/latex] and [latex]g(x)=9[\/latex]. However, when we are solving an equation algebraically, there is no function in sight so we can just report the value of the variable (e.g., [latex]x[\/latex]).<\/p>\n<h2>Solving Exponential Equations Using the Property of Equality<\/h2>\n<p>We may use the <em><strong>property of equality<\/strong><\/em> for solving an exponential equation if the exponential expressions on each side of the equation have the same base.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: left;\">Property of Equality<\/h3>\n<p style=\"text-align: center;\">For any real numbers [latex]a, \\;x[\/latex] and [latex]y[\/latex],<\/p>\n<p style=\"text-align: center;\">If [latex]a^x = a^y[\/latex], then [latex]x = y[\/latex].<\/p>\n<\/div>\n<p>For example, to solve the equation [latex]3^x = 9[\/latex], we can write [latex]9 = 3^2[\/latex] so that [latex]3^x = 3^2[\/latex]. The property of equality then tells us that [latex]x = 2[\/latex] because with the same base, the exponents must be equal. Although it may at first appear that the exponential expressions in an equation do not have the same base, they can often be transformed to expressions with the same base.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Solve the equation [latex]32\\left(2^{x+3}\\right)=64[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}32\\left(2^{x+3}\\right)&=64&&\\text{The base of the exponential is 2, so we try to write 32 as a power of 2}\\\\2^5\\times 2^{x+3}&=2^6&&\\text{Product rule}\\\\2^{x+3+5}&=2^6\\\\2^{x+8}&=2^6&&\\text{Now each side has the same base}\\\\x+8&=6&&\\text{The exponents must be equal}\\\\x&=-2\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: center;\">\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Solve the equation [latex]32\\left(2^{x+3}\\right)=64[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Another, and more efficient way to solve the same equation as example 1, is to notice that each side of the equation is divisible by 32:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}32\\left(2^{x+3}\\right)&=64&&\\text{Divide each side by 32}\\\\2^{x+3}&=2&&\\text{Same base so exponents must be equal}\\\\x+3&=1&&\\text{Subtract 3 from both sides}\\\\x&=-2\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>TRY IT 1<\/h3>\n<p>Solve the equation [latex]15\\left(3^{x+4}\\right)=45[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm577\">Show Answer<\/span><\/p>\n<div id=\"qhjm577\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Solve the equation [latex]\\dfrac{1}{4}\\left(6^{2x-1}\\right)=9[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm397\">Show Answer<\/span><\/p>\n<div id=\"qhjm397\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=\\dfrac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Solve the equation [latex]27^{x+1}=81^{x-2}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The bases of the exponential expressions are 27 and 81. Both of these can be written as a power of 3: [latex]27 = 3^3[\/latex] and\u00a0[latex]81 = 3^4[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}27^{x+1}&=81^{x-2}\\\\{\\left(3^3\\right)}^{x+1}&={\\left(3^4\\right)}^{x-2}&&\\text{Power to a power rule}\\\\3^{3(x+1)}&=3^{4(x-2)}&&\\text{Same base so exponents must be equal}\\\\3(x+1)&=4(x-2)&&\\text{Distribute}\\\\3x+3&=4x-8&&\\text{Subtract }3x\\text{ from both sides}\\\\3&=x-8&&\\text{Add 8 to both sides}\\\\11&=x\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Solve the equation\u00a0[latex]2^{-x+5}=16^{x+1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qLeo410\">Show Answer<\/span><\/p>\n<div id=\"qLeo410\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=\\dfrac{1}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Determine the intersection point of the functions [latex]f(x)=5\\left(2^{x-3}\\right)[\/latex] and [latex]g(x)=10\\left(2^{-x+16}\\right)[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: left;\">The intersection point occurs at an [latex]x[\/latex]-value where [latex]f(x)=g(x)[\/latex].<\/p>\n<p>Set\u00a0[latex]f(x)=g(x)[\/latex] and solve for [latex]x[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}5\\left(2^{x-3}\\right)&=10\\left(2^{-x+16}\\right)\\\\2^{x-3}&=2\\left(2^{-x+16}\\right)\\\\2^{x-3}&=2^{-x+17}\\\\x-3&=-x+17\\\\2x-3&=17\\\\2x&=20\\\\x&=10\\end{aligned}[\/latex]<\/p>\n<p>Now that we know the value of [latex]x[\/latex], we can find the corresponding value of [latex]y[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}y&=f(10)\\\\&=5\\left(2^{10-3}\\right)\\\\&=5\\left(2^7\\right)\\\\&=5(128)\\\\&=640\\end{aligned}[\/latex]<\/p>\n<p>Consequently, the intersection point is [latex](10, 640)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Note: We could also have used [latex]g(x)[\/latex] to find the value of [latex]y[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}y&=g(10)\\\\&=10\\left(2^{-10+16}\\right)\\\\&=10\\left(2^{6}\\right)\\\\&=10\\cdot 64\\\\&=640\\end{aligned}[\/latex]<\/p>\n<p>This is a good check for the answer.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Determine the intersection point of the functions [latex]f(x)=6\\left(3^x\\right)[\/latex] and [latex]g(x)=18\\left(3^{-x+1}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm201\">Show Answer<\/span><\/p>\n<div id=\"qhjm201\" class=\"hidden-answer\" style=\"display: none\">\n<p>(1, 18)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-2237\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Solve Exponential Equations. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Examples 2, 3, 4; Try it: hjm397; hjm577; hjm201. <strong>Authored by<\/strong>: Hazel McKenna . <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Try it: Leo410. <strong>Authored by<\/strong>: Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":422608,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Solve Exponential Equations\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Examples 2, 3, 4; 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