{"id":11314,"date":"2015-07-14T19:40:39","date_gmt":"2015-07-14T19:40:39","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&#038;p=11314"},"modified":"2021-12-29T19:16:54","modified_gmt":"2021-12-29T19:16:54","slug":"evaluating-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/evaluating-logarithms\/","title":{"raw":"Evaluating logarithms","rendered":"Evaluating logarithms"},"content":{"raw":"<section id=\"fs-id1165137530906\" data-depth=\"1\">\r\n<p id=\"fs-id1165137422589\">Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\\mathrm{log}}_{2}8[\/latex]. We ask, \"To what exponent must 2\u00a0be raised in order to get 8?\" Because we already know [latex]{2}^{3}=8[\/latex], it follows that [latex]{\\mathrm{log}}_{2}8=3[\/latex].<\/p>\r\n<p id=\"fs-id1165137733822\">Now consider solving [latex]{\\mathrm{log}}_{7}49[\/latex] and [latex]{\\mathrm{log}}_{3}27[\/latex] mentally.<\/p>\r\n\r\n<ul id=\"fs-id1165137937690\">\r\n \t<li>We ask, \"To what exponent must 7 be raised in order to get 49?\" We know [latex]{7}^{2}=49[\/latex]. Therefore, [latex]{\\mathrm{log}}_{7}49=2[\/latex]<\/li>\r\n \t<li>We ask, \"To what exponent must 3 be raised in order to get 27?\" We know [latex]{3}^{3}=27[\/latex]. Therefore, [latex]{\\mathrm{log}}_{3}27=3[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137456358\">Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.<\/p>\r\n\r\n<ul id=\"fs-id1165137584208\">\r\n \t<li>We ask, \"To what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}[\/latex]? \" We know [latex]{2}^{2}=4[\/latex] and [latex]{3}^{2}=9[\/latex], so [latex]{\\left(\\frac{2}{3}\\right)}^{2}=\\frac{4}{9}[\/latex]. Therefore, [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2[\/latex].<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1165137455840\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\r\n<h3>How To: evaluate a logarithm without the calculator.<\/h3>\r\n<ol id=\"fs-id1165134079724\" data-number-style=\"arabic\">\r\n \t<li>Rewrite the logarithm in exponential form: [latex]{b}^{y}=x[\/latex].<\/li>\r\n \t<li>Use previous knowledge of powers of <em>b<\/em>\u00a0to identify <em>y<\/em>\u00a0by asking, \"To what exponent should <em>b<\/em>\u00a0be raised in order to get <em>x<\/em>?\"<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.\r\n\r\n[reveal-answer q=\"293311\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"293311\"]\r\n\r\nFirst we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[\/latex].\r\n\r\nNext, we ask, \"To what exponent must 4 be raised in order to get 64?\"\r\n\r\nWe know [latex]{4}^{3}=64[\/latex].\r\n\r\nTherefore, [latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137745041\">Solve [latex]y={\\mathrm{log}}_{121}\\left(11\\right)[\/latex] without using a calculator.<\/p>\r\n[reveal-answer q=\"864518\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"864518\"]\r\n\r\n[latex]\\mathrm{log}_{121}\u200b (11)= \u200b\\frac{1}{2}\u200b\u200b \\;(\\text{recalling that} \\displaystyle \\sqrt{121}={\\left(121\\right)}^{\\frac{1}{2}}=11)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex] without using a calculator.\r\n\r\n[reveal-answer q=\"522177\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"522177\"]\r\n<p id=\"fs-id1165137638179\">First we rewrite the logarithm in exponential form: [latex]{3}^{y}=\\frac{1}{27}[\/latex]. Next, we ask, \"To what exponent must 3 be raised in order to get [latex]\\frac{1}{27}[\/latex]\"?<\/p>\r\n<p id=\"fs-id1165137552085\">We know [latex]{3}^{3}=27[\/latex], but what must we do to get the reciprocal, [latex]\\frac{1}{27}[\/latex]? Recall from working with exponents that [latex]{b}^{-a}=\\frac{1}{{b}^{a}}[\/latex]. We use this information to write<\/p>\r\n\r\n<div id=\"eip-id1165137550550\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}{3}^{-3}=\\frac{1}{{3}^{3}}\\hfill \\\\ =\\frac{1}{27}\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137585807\">Therefore, [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135437134\">Evaluate [latex]y={\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)[\/latex] without using a calculator.<\/p>\r\n[reveal-answer q=\"925731\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"925731\"]\\displaystyle {\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)=-5[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>When we talked about exponential functions, we introduced the number\u00a0<em data-effect=\"italics\">e<\/em>. Just as\u00a0<em data-effect=\"italics\">e<\/em>\u00a0was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base\u00a0<em data-effect=\"italics\">e<\/em>\u00a0is called the\u00a0<strong><span id=\"afd63274-5172-405b-b787-bfd578b98d44_term293\" data-type=\"term\">natural logarithmic function<\/span><\/strong>. The function <span style=\"white-space: nowrap;\">[latex]f(x)=log_e{x}[\/latex]<\/span>\u00a0is generally written [latex]f(x)=ln x[\/latex] and we read it as \u201cel en of\u00a0<span class=\"os-math-in-para\"><span id=\"MathJax-Element-210-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mo&gt;&amp;#x201D;&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mo&gt;\u201d&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-2783\" class=\"math\"><span id=\"MathJax-Span-2784\" class=\"mrow\"><span id=\"MathJax-Span-2785\" class=\"semantics\"><span id=\"MathJax-Span-2786\" class=\"mrow\"><span id=\"MathJax-Span-2787\" class=\"mrow\"><span id=\"MathJax-Span-2788\" class=\"mi\">\ud835\udc65<\/span><span id=\"MathJax-Span-2789\" class=\"mo\">.<\/span><span id=\"MathJax-Span-2790\" class=\"mo\">\u201d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nWhen the base of the logarithm function is 10, we call it the\u00a0<strong><span id=\"afd63274-5172-405b-b787-bfd578b98d44_term294\" data-type=\"term\">common logarithmic function<\/span><\/strong>\u00a0and the base is not shown. If the base\u00a0<em data-effect=\"italics\">a<\/em>\u00a0of a logarithm is not shown, as in\u00a0<span style=\"white-space: nowrap;\">[latex]f(x)=log{x}[\/latex]<\/span>, we assume the base is 10.\r\n\r\nMost values of [latex]\\mathrm{ln}\\left(x\\right)[\/latex] can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, [latex]\\mathrm{ln}1=0[\/latex]. For other natural logarithms, we can use the [latex]\\mathrm{ln}[\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of <em>e<\/em>\u00a0using the inverse property of logarithms.\r\n\r\n<img src=\"https:\/\/openstax.org\/apps\/archive\/20210823.155019\/resources\/2efc522670c76615bdf03a52dd94284b3f94373b\" alt=\"It will be important for you to use your calculator to evaluate both common and natural logarithms. Find the log and ln keys on your calculator.\" \/>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137435623\">Evaluate [latex]\\mathrm{ln}\\left(500\\right)[\/latex].<\/p>\r\n[reveal-answer q=\"677894\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"677894\"]\r\n<ul>\r\n \t<li>Press [LN].<\/li>\r\n \t<li>Enter 500, followed by [ ) ].<\/li>\r\n \t<li>Press [ENTER].<\/li>\r\n<\/ul>\r\nRounding to four decimal places, [latex]\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video gives more examples of converting between exponential and logarithmic form as well as evaluating logarithms.\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=z296tOPj0HA[\/embed]","rendered":"<section id=\"fs-id1165137530906\" data-depth=\"1\">\n<p id=\"fs-id1165137422589\">Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\\mathrm{log}}_{2}8[\/latex]. We ask, &#8220;To what exponent must 2\u00a0be raised in order to get 8?&#8221; Because we already know [latex]{2}^{3}=8[\/latex], it follows that [latex]{\\mathrm{log}}_{2}8=3[\/latex].<\/p>\n<p id=\"fs-id1165137733822\">Now consider solving [latex]{\\mathrm{log}}_{7}49[\/latex] and [latex]{\\mathrm{log}}_{3}27[\/latex] mentally.<\/p>\n<ul id=\"fs-id1165137937690\">\n<li>We ask, &#8220;To what exponent must 7 be raised in order to get 49?&#8221; We know [latex]{7}^{2}=49[\/latex]. Therefore, [latex]{\\mathrm{log}}_{7}49=2[\/latex]<\/li>\n<li>We ask, &#8220;To what exponent must 3 be raised in order to get 27?&#8221; We know [latex]{3}^{3}=27[\/latex]. Therefore, [latex]{\\mathrm{log}}_{3}27=3[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137456358\">Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.<\/p>\n<ul id=\"fs-id1165137584208\">\n<li>We ask, &#8220;To what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}[\/latex]? &#8221; We know [latex]{2}^{2}=4[\/latex] and [latex]{3}^{2}=9[\/latex], so [latex]{\\left(\\frac{2}{3}\\right)}^{2}=\\frac{4}{9}[\/latex]. Therefore, [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2[\/latex].<\/li>\n<\/ul>\n<div id=\"fs-id1165137455840\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3>How To: evaluate a logarithm without the calculator.<\/h3>\n<ol id=\"fs-id1165134079724\" data-number-style=\"arabic\">\n<li>Rewrite the logarithm in exponential form: [latex]{b}^{y}=x[\/latex].<\/li>\n<li>Use previous knowledge of powers of <em>b<\/em>\u00a0to identify <em>y<\/em>\u00a0by asking, &#8220;To what exponent should <em>b<\/em>\u00a0be raised in order to get <em>x<\/em>?&#8221;<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q293311\">Show Answer<\/span><\/p>\n<div id=\"q293311\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[\/latex].<\/p>\n<p>Next, we ask, &#8220;To what exponent must 4 be raised in order to get 64?&#8221;<\/p>\n<p>We know [latex]{4}^{3}=64[\/latex].<\/p>\n<p>Therefore, [latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137745041\">Solve [latex]y={\\mathrm{log}}_{121}\\left(11\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q864518\">Show Answer<\/span><\/p>\n<div id=\"q864518\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\mathrm{log}_{121}\u200b (11)= \u200b\\frac{1}{2}\u200b\u200b \\;(\\text{recalling that} \\displaystyle \\sqrt{121}={\\left(121\\right)}^{\\frac{1}{2}}=11)[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q522177\">Show Answer<\/span><\/p>\n<div id=\"q522177\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137638179\">First we rewrite the logarithm in exponential form: [latex]{3}^{y}=\\frac{1}{27}[\/latex]. Next, we ask, &#8220;To what exponent must 3 be raised in order to get [latex]\\frac{1}{27}[\/latex]&#8220;?<\/p>\n<p id=\"fs-id1165137552085\">We know [latex]{3}^{3}=27[\/latex], but what must we do to get the reciprocal, [latex]\\frac{1}{27}[\/latex]? Recall from working with exponents that [latex]{b}^{-a}=\\frac{1}{{b}^{a}}[\/latex]. We use this information to write<\/p>\n<div id=\"eip-id1165137550550\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}{3}^{-3}=\\frac{1}{{3}^{3}}\\hfill \\\\ =\\frac{1}{27}\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137585807\">Therefore, [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135437134\">Evaluate [latex]y={\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q925731\">Show Answer<\/span><\/p>\n<div id=\"q925731\" class=\"hidden-answer\" style=\"display: none\">\\displaystyle {\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)=-5<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>When we talked about exponential functions, we introduced the number\u00a0<em data-effect=\"italics\">e<\/em>. Just as\u00a0<em data-effect=\"italics\">e<\/em>\u00a0was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base\u00a0<em data-effect=\"italics\">e<\/em>\u00a0is called the\u00a0<strong><span id=\"afd63274-5172-405b-b787-bfd578b98d44_term293\" data-type=\"term\">natural logarithmic function<\/span><\/strong>. The function <span style=\"white-space: nowrap;\">[latex]f(x)=log_e{x}[\/latex]<\/span>\u00a0is generally written [latex]f(x)=ln x[\/latex] and we read it as \u201cel en of\u00a0<span class=\"os-math-in-para\"><span id=\"MathJax-Element-210-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mo&gt;&amp;#x201D;&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mo&gt;\u201d&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-2783\" class=\"math\"><span id=\"MathJax-Span-2784\" class=\"mrow\"><span id=\"MathJax-Span-2785\" class=\"semantics\"><span id=\"MathJax-Span-2786\" class=\"mrow\"><span id=\"MathJax-Span-2787\" class=\"mrow\"><span id=\"MathJax-Span-2788\" class=\"mi\">\ud835\udc65<\/span><span id=\"MathJax-Span-2789\" class=\"mo\">.<\/span><span id=\"MathJax-Span-2790\" class=\"mo\">\u201d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>When the base of the logarithm function is 10, we call it the\u00a0<strong><span id=\"afd63274-5172-405b-b787-bfd578b98d44_term294\" data-type=\"term\">common logarithmic function<\/span><\/strong>\u00a0and the base is not shown. If the base\u00a0<em data-effect=\"italics\">a<\/em>\u00a0of a logarithm is not shown, as in\u00a0<span style=\"white-space: nowrap;\">[latex]f(x)=log{x}[\/latex]<\/span>, we assume the base is 10.<\/p>\n<p>Most values of [latex]\\mathrm{ln}\\left(x\\right)[\/latex] can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, [latex]\\mathrm{ln}1=0[\/latex]. For other natural logarithms, we can use the [latex]\\mathrm{ln}[\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of <em>e<\/em>\u00a0using the inverse property of logarithms.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/openstax.org\/apps\/archive\/20210823.155019\/resources\/2efc522670c76615bdf03a52dd94284b3f94373b\" alt=\"It will be important for you to use your calculator to evaluate both common and natural logarithms. Find the log and ln keys on your calculator.\" \/><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137435623\">Evaluate [latex]\\mathrm{ln}\\left(500\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q677894\">Solution<\/span><\/p>\n<div id=\"q677894\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Press [LN].<\/li>\n<li>Enter 500, followed by [ ) ].<\/li>\n<li>Press [ENTER].<\/li>\n<\/ul>\n<p>Rounding to four decimal places, [latex]\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>The following video gives more examples of converting between exponential and logarithmic form as well as evaluating logarithms.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Introduction to Logarithms\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/z296tOPj0HA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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-11314\">\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":276,"menu_order":13,"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":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-11314","chapter","type-chapter","status-publish","hentry"],"part":11277,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11314","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11314\/revisions"}],"predecessor-version":[{"id":16394,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11314\/revisions\/16394"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/11277"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11314\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=11314"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=11314"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=11314"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=11314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}