{"id":233,"date":"2023-11-08T16:10:33","date_gmt":"2023-11-08T16:10:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/use-compound-interest-formulas\/"},"modified":"2024-08-13T22:43:08","modified_gmt":"2024-08-13T22:43:08","slug":"7-6-solving-exponential-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/7-6-solving-exponential-equations\/","title":{"raw":"7.6 Solving Exponential Equations","rendered":"7.6 Solving Exponential Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve exponential equations with common bases.<\/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 is a consequence of the <strong>One-to-One Property of Exponents<\/strong>, which previously allowed us to define its inverse function, the logarithm.<\/p>\r\nAn example of such an equation is [latex]2^x=2^5[\/latex]. Since the exponential function is one-to-one, the exponent on the left must be [latex]5[\/latex] in order for the two sides to equal. No other exponent will result in the same value on both sides.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>ONE-TO-ONE PROPERTY OF EXPONENTS<\/h3>\r\nLet [latex]b&gt;0[\/latex] and [latex]b \\neq 1.[\/latex] Then [latex]b^x = b^y[\/latex] implies [latex]x=y.[\/latex]\r\n\r\nThis also applies when the exponents are algebraic expressions.\r\n\r\n<\/div>\r\nTherefore, 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.\r\n<p id=\"fs-id1165135192889\">Consider the equation [latex]{3}^{4x - 7}=3^{2x-1}[\/latex]. To solve for [latex]x[\/latex], we apply the one-to-one property of exponents by setting the exponents equal to one another:<\/p>\r\n\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}3^{4x - 7} &amp;=3^{2x - 1} \\\\ 4x - 7 &amp;=2x - 1&amp;&amp; \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\ 2x &amp;=6 &amp;&amp; \\color{blue}{\\textsf{subtract 2}x\\textsf{ and add 7 to both sides}}\\\\ x&amp; =3 &amp;&amp; \\color{blue}{\\textsf{divide by 2}}\\end{align}[\/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<p style=\"text-align: center;\">[latex]\\begin{align} 2^{x - 1}&amp;=2^{2x - 4}&amp;&amp; \\color{blue}{\\textsf{The common base is }2}\\\\ x - 1&amp;=2x - 4 &amp;&amp; \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\ x&amp;=3 &amp;&amp; \\color{blue}{\\textsf{solve for }x}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\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 if the bases of an exponential equation are not equal, we can rewrite the terms as powers with a common base and solve using the One-to-One Property of Exponents. For 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] or\u00a0[latex]2^{-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 of Exponents, to solve for [latex]x[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}256&amp;={4}^{x - 5} \\\\\r\n2^8&amp;=\\left(2^2\\right)^{x-5} &amp;&amp; \\color{blue}{\\textsf{rewrite the base on each side as a power of }2} \\\\\r\n2^8&amp;=2^{2\\cdot(x-5)} &amp;&amp; \\color{blue}{\\textsf{use the Power Rule for Exponents}} \\\\\r\n8&amp;=2x-10 &amp;&amp; \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\\r\n18&amp;=2x \\\\\r\n9&amp;=x \\end{align}[\/latex]<\/p>\r\n\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">The next example is similar but both exponents contain variables.<\/div>\r\n<div><\/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\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}8^{x+2}&amp;=16^{x+1} \\\\\r\n\\left(2^3\\right)^{x+2}&amp;=\\left(2^4\\right)^{x+1} &amp;&amp; \\color{blue}{\\textsf{rewrite the base on each side as a power of }2} \\\\\r\n2^{3x+6}&amp;=2^{4x+1} &amp;&amp; \\color{blue}{\\textsf{use the Power Rule for Exponents}} \\\\\r\n3x+6&amp;=4x+4 &amp;&amp; \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\\r\n2&amp;=x \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRemember that you can write radicals as rational exponents, so you may be able to find common bases when it is not completely obvious at first.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]{5}^{7x}=\\sqrt{5}[\/latex].\r\n[reveal-answer q=\"507738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507738\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}5^{7x}&amp;=\\sqrt{5} \\\\\r\n5^{7x}&amp;=5^{1\/2} &amp;&amp; \\color{blue}{\\textsf{rewrite the right side as a rational exponent}} \\\\\r\n7x&amp;=\\dfrac{1}{2} &amp;&amp; \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\\r\nx&amp;=\\dfrac{1}{14} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\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\r\nBefore revealing the answer to the example, think about the range of the exponential function, which is the set of output values it is allowed to produce.\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\nThe range of the exponential function [latex]f(x)=3^{x+1}[\/latex] is the interval [latex](0,\\infty).[\/latex] Since [latex]-2[\/latex] is not in this interval, it is impossible for any value of [latex]x[\/latex] to make the left side equal\u00a0[latex]-2.[\/latex] This equation has no solution.\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 since the exponential function has a horizontal asymptote of [latex]y=0,[\/latex] 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 by setting the exponents equal to each other. The 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 the exponential function can only produce positive values as outputs.\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve exponential equations with common bases.<\/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 is a consequence of the <strong>One-to-One Property of Exponents<\/strong>, which previously allowed us to define its inverse function, the logarithm.<\/p>\n<p>An example of such an equation is [latex]2^x=2^5[\/latex]. Since the exponential function is one-to-one, the exponent on the left must be [latex]5[\/latex] in order for the two sides to equal. No other exponent will result in the same value on both sides.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>ONE-TO-ONE PROPERTY OF EXPONENTS<\/h3>\n<p>Let [latex]b>0[\/latex] and [latex]b \\neq 1.[\/latex] Then [latex]b^x = b^y[\/latex] implies [latex]x=y.[\/latex]<\/p>\n<p>This also applies when the exponents are algebraic expressions.<\/p>\n<\/div>\n<p>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\">Consider the equation [latex]{3}^{4x - 7}=3^{2x-1}[\/latex]. To solve for [latex]x[\/latex], we apply the one-to-one property of exponents by setting the exponents equal to one another:<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}3^{4x - 7} &=3^{2x - 1} \\\\ 4x - 7 &=2x - 1&& \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\ 2x &=6 && \\color{blue}{\\textsf{subtract 2}x\\textsf{ and add 7 to both sides}}\\\\ x& =3 && \\color{blue}{\\textsf{divide by 2}}\\end{align}[\/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 style=\"text-align: center;\">[latex]\\begin{align} 2^{x - 1}&=2^{2x - 4}&& \\color{blue}{\\textsf{The common base is }2}\\\\ x - 1&=2x - 4 && \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\ x&=3 && \\color{blue}{\\textsf{solve for }x}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\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 if the bases of an exponential equation are not equal, we can rewrite the terms as powers with a common base and solve using the One-to-One Property of Exponents. For 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] or\u00a0[latex]2^{-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 of Exponents, to solve for [latex]x[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}256&={4}^{x - 5} \\\\  2^8&=\\left(2^2\\right)^{x-5} && \\color{blue}{\\textsf{rewrite the base on each side as a power of }2} \\\\  2^8&=2^{2\\cdot(x-5)} && \\color{blue}{\\textsf{use the Power Rule for Exponents}} \\\\  8&=2x-10 && \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\  18&=2x \\\\  9&=x \\end{align}[\/latex]<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">The next example is similar but both exponents contain variables.<\/div>\n<div><\/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\">\n<p style=\"text-align: center;\">[latex]\\begin{align}8^{x+2}&=16^{x+1} \\\\  \\left(2^3\\right)^{x+2}&=\\left(2^4\\right)^{x+1} && \\color{blue}{\\textsf{rewrite the base on each side as a power of }2} \\\\  2^{3x+6}&=2^{4x+1} && \\color{blue}{\\textsf{use the Power Rule for Exponents}} \\\\  3x+6&=4x+4 && \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\  2&=x \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Remember that you can write radicals as rational exponents, so you may be able to find common bases when it is not completely obvious at first.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]{5}^{7x}=\\sqrt{5}[\/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 style=\"text-align: center;\">[latex]\\begin{align}5^{7x}&=\\sqrt{5} \\\\  5^{7x}&=5^{1\/2} && \\color{blue}{\\textsf{rewrite the right side as a rational exponent}} \\\\  7x&=\\dfrac{1}{2} && \\color{blue}{\\textsf{apply the One-to-One Property of Exponents}} \\\\  x&=\\dfrac{1}{14} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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<p>Before revealing the answer to the example, think about the range of the exponential function, which is the set of output values it is allowed to produce.<\/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>The range of the exponential function [latex]f(x)=3^{x+1}[\/latex] is the interval [latex](0,\\infty).[\/latex] Since [latex]-2[\/latex] is not in this interval, it is impossible for any value of [latex]x[\/latex] to make the left side equal\u00a0[latex]-2.[\/latex] This equation has no solution.<\/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 since the exponential function has a horizontal asymptote of [latex]y=0,[\/latex] 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 by setting the exponents equal to each other. The 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 the exponential function can only produce positive values as outputs.<\/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-233\">\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":395986,"menu_order":6,"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":"c5bacacb-c0e9-4576-aa6f-d4501b15e277","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-233","chapter","type-chapter","status-publish","hentry"],"part":225,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/233","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/233\/revisions"}],"predecessor-version":[{"id":1780,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/233\/revisions\/1780"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/parts\/225"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/233\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/media?parent=233"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=233"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/contributor?post=233"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/license?post=233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}