{"id":1458,"date":"2021-08-24T19:11:47","date_gmt":"2021-08-24T19:11:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1458"},"modified":"2022-01-31T21:48:12","modified_gmt":"2022-01-31T21:48:12","slug":"exponential-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/exponential-equations\/","title":{"raw":"Exponential Equations","rendered":"Exponential Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Apply the one-to-one property of exponents to solve an exponential equation<\/li>\r\n \t<li>Solve exponential equations of the form [latex]y=Ae^{kt} \\ \\mathrm{for} \\ t[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165135192781\">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. We can set the exponents equal to one another and solve for the unknown.<\/p>\r\nFor example, to solve the exponential equation [latex]2^{x} = 8[\/latex], we might note that [latex]8[\/latex] can be written as [latex]2^{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex]2^x=8[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]2^x=2^3[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=3[\/latex]<\/p>\r\nIn general, we can summarize solving exponential equations whose terms all have the same base in this way:\r\n\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 class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]3^{x-1}=81[\/latex]\r\n\r\n[reveal-answer q=\"96632\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"96632\"]\r\n\r\nWrite the two sides of the equation as exponential expressions with the same base.\r\n<p style=\"text-align: center;\">[latex]3^{x-1}=3^{4}[\/latex]<\/p>\r\nUse the one-to-one property to set the exponents equal to each other.\r\n<p style=\"text-align: center;\">[latex]x-1=4[\/latex]<\/p>\r\nSolve the resulting equation for the unknown.\r\n<p style=\"text-align: center;\">[latex]x=4+1[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]217502[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Exponential Equations with Unlike Bases<\/h2>\r\nSometimes it is not possible to write both sides of an exponential equation as powers of the same base. Base [latex]e[\/latex] is a very common base found in science, finance, and engineering applications. When we have an equation with a base [latex]e[\/latex] on either side, we can use the natural logarithm to solve it. Earlier, we introduced a formula that models continuous growth\/decay, [latex]y=Ae^{kt}[\/latex]. To solve this equation for [latex]t[\/latex],\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Isolate the exponential expression.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Use the natural logarithm function to write the exponential equation in logarithmic form.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Solve for [latex]t[\/latex].<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]300=5e^{2t}[\/latex].\r\n\r\n[reveal-answer q=\"298549\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"298549\"]\r\n<p style=\"text-align: center;\">[latex]300=5e^{2t}[\/latex]<\/p>\r\nIsolate the exponential.\r\n<p style=\"text-align: center;\">[latex]\\frac{300}{5}=e^{2t}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]6=e^{2t}[\/latex]<\/p>\r\nConvert to logarithmic form.\r\n<p style=\"text-align: center;\">[latex]2t= \\mathrm{ln} \\ (6)[\/latex]<\/p>\r\nSolve for [latex]t[\/latex].\r\n<p style=\"text-align: center;\">[latex]t=\\frac{\\mathrm{ln} \\ (6)}{2} \\approx 0.8959[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]0.6=0.1+e^{\\frac{t}{2}}[\/latex].\r\n\r\n[reveal-answer q=\"431922\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"431922\"]\r\n<p style=\"text-align: center;\">[latex]0.6=0.1+e^{\\frac{t}{2}}[\/latex]<\/p>\r\nIsolate the exponential.\r\n<p style=\"text-align: center;\">[latex]0.6-0.1=e^{\\frac{t}{2}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]0.5=e^{\\frac{t}{2}}[\/latex]<\/p>\r\nConvert to logarithmic form.\r\n<p style=\"text-align: center;\">[latex]\\frac{t}{2}= \\mathrm{ln} \\ (0.5)[\/latex]<\/p>\r\nSolve for [latex]t[\/latex].\r\n<p style=\"text-align: center;\">[latex]t=2 \\ \\mathrm{ln} \\ (0.5) \\approx -1.3863[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSometimes an equation may not have a solution. For example, to solve [latex]e^{t}= -2[\/latex] we might convert to logarithmic form to obtain [latex]t=\\mathrm{ln} \\ (-2)[\/latex]. Entering this on your calculator produces an error message. We can\u2019t take a logarithm of a negative number. The original expression, [latex]e^{t}= -2[\/latex] is never true since [latex]e^{t}&gt;0[\/latex] for all real numbers [latex]t[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]1536[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Apply the one-to-one property of exponents to solve an exponential equation<\/li>\n<li>Solve exponential equations of the form [latex]y=Ae^{kt} \\ \\mathrm{for} \\ t[\/latex]<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135192781\">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. We can set the exponents equal to one another and solve for the unknown.<\/p>\n<p>For example, to solve the exponential equation [latex]2^{x} = 8[\/latex], we might note that [latex]8[\/latex] can be written as [latex]2^{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]2^x=8[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]2^x=2^3[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=3[\/latex]<\/p>\n<p>In general, we can summarize solving exponential equations whose terms all have the same base in this way:<\/p>\n<p>For 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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]3^{x-1}=81[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96632\">Show Answer<\/span><\/p>\n<div id=\"q96632\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write the two sides of the equation as exponential expressions with the same base.<\/p>\n<p style=\"text-align: center;\">[latex]3^{x-1}=3^{4}[\/latex]<\/p>\n<p>Use the one-to-one property to set the exponents equal to each other.<\/p>\n<p style=\"text-align: center;\">[latex]x-1=4[\/latex]<\/p>\n<p>Solve the resulting equation for the unknown.<\/p>\n<p style=\"text-align: center;\">[latex]x=4+1[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm217502\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=217502&theme=oea&iframe_resize_id=ohm217502&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Exponential Equations with Unlike Bases<\/h2>\n<p>Sometimes it is not possible to write both sides of an exponential equation as powers of the same base. Base [latex]e[\/latex] is a very common base found in science, finance, and engineering applications. When we have an equation with a base [latex]e[\/latex] on either side, we can use the natural logarithm to solve it. Earlier, we introduced a formula that models continuous growth\/decay, [latex]y=Ae^{kt}[\/latex]. To solve this equation for [latex]t[\/latex],<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Isolate the exponential expression.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Use the natural logarithm function to write the exponential equation in logarithmic form.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Solve for [latex]t[\/latex].<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]300=5e^{2t}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q298549\">Show Answer<\/span><\/p>\n<div id=\"q298549\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]300=5e^{2t}[\/latex]<\/p>\n<p>Isolate the exponential.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{300}{5}=e^{2t}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]6=e^{2t}[\/latex]<\/p>\n<p>Convert to logarithmic form.<\/p>\n<p style=\"text-align: center;\">[latex]2t= \\mathrm{ln} \\ (6)[\/latex]<\/p>\n<p>Solve for [latex]t[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]t=\\frac{\\mathrm{ln} \\ (6)}{2} \\approx 0.8959[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]0.6=0.1+e^{\\frac{t}{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q431922\">Show Answer<\/span><\/p>\n<div id=\"q431922\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]0.6=0.1+e^{\\frac{t}{2}}[\/latex]<\/p>\n<p>Isolate the exponential.<\/p>\n<p style=\"text-align: center;\">[latex]0.6-0.1=e^{\\frac{t}{2}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]0.5=e^{\\frac{t}{2}}[\/latex]<\/p>\n<p>Convert to logarithmic form.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{t}{2}= \\mathrm{ln} \\ (0.5)[\/latex]<\/p>\n<p>Solve for [latex]t[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]t=2 \\ \\mathrm{ln} \\ (0.5) \\approx -1.3863[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Sometimes an equation may not have a solution. For example, to solve [latex]e^{t}= -2[\/latex] we might convert to logarithmic form to obtain [latex]t=\\mathrm{ln} \\ (-2)[\/latex]. Entering this on your calculator produces an error message. We can\u2019t take a logarithm of a negative number. The original expression, [latex]e^{t}= -2[\/latex] is never true since [latex]e^{t}>0[\/latex] for all real numbers [latex]t[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1536\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1536&theme=oea&iframe_resize_id=ohm1536&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-1458\">\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><strong>Provided by<\/strong>: Lumen Learning. <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 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=\"https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions\">https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions<\/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>: Access for free at https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions<\/li><li>QID 217502. <strong>Authored by<\/strong>: Daniel Breuer. <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>: IMathAS Community License CC-BY + GPL<\/li><li>QID 1536. <strong>Authored by<\/strong>: David Lippman. <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>: IMathAS Community License CC-BY + GPL<\/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":169134,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions\"},{\"type\":\"cc\",\"description\":\"QID 217502\",\"author\":\"Daniel Breuer\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"QID 1536\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1458","chapter","type-chapter","status-publish","hentry"],"part":249,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1458","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1458\/revisions"}],"predecessor-version":[{"id":3604,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1458\/revisions\/3604"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/249"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1458\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1458"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1458"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1458"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1458"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}