{"id":1179,"date":"2024-04-15T14:28:55","date_gmt":"2024-04-15T14:28:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1179"},"modified":"2024-08-04T12:18:17","modified_gmt":"2024-08-04T12:18:17","slug":"exponential-and-logarithmic-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/exponential-and-logarithmic-functions-fresh-take\/","title":{"raw":"Exponential and Logarithmic Functions: Fresh Take","rendered":"Exponential and Logarithmic Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Identify and evaluate exponential functions<\/li>\r\n\t<li>Analyze logarithmic functions by identifying forms, understanding their exponential relationships, and calculating different base logarithms<\/li>\r\n\t<li>Identify the hyperbolic functions, their graphs, and basic identities\u00a0<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Exponential Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Definition of Exponential Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Form: [latex]f(x) = ab^x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is a non-zero real number (initial value)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]b[\/latex] is a positive real number, [latex]b \\neq 1[\/latex] (base)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Characteristics:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Constant base, variable exponent<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Rapid growth compared to power functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Domain: all real numbers<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Range: [latex](0, \u221e)[\/latex] for [latex]b &gt; 1[\/latex], [latex](0, \u221e)[\/latex] for [latex]0 &lt; b &lt; 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Evaluating Exponential Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Substitute the given value for [latex]x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Follow order of operations<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to parentheses and exponents<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Laws of Exponents:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Product of Powers: [latex]b^x \\cdot b^y = b^{x+y}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Quotient of Powers: [latex]\\frac{b^x}{b^y} = b^{x-y}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Power of a Power: [latex](b^x)^y=b^{xy}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Power of a Product: [latex](ab)^x=a^x b^x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Power of a Quotient: [latex]\\dfrac{a^x}{b^x} =\\left(\\dfrac{a}{b}\\right)^x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applications:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Population growth<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compound interest<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Radioactive decay<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572173715\">Given the exponential function [latex]f(x)=100\u00b73^{x\/2}[\/latex], evaluate [latex]f(4)[\/latex] and [latex]f(10)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572173781\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572173781\"]<\/p>\r\n<p id=\"fs-id1170572173781\">[latex]f(4)=900; \\, f(10)=24,300[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox interact\">\r\n<p><a href=\"https:\/\/www.worldpopulationbalance.org\/understanding-exponential-growth\" target=\"_blank\" rel=\"noopener\">Go to World Population Balance for another example of exponential population growth.<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Simplify the following expression and then evaluate it for [latex]x = 2[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\dfrac{(3x^{2})^3 \\cdot 2^{x-1}}{9x^{-1} \\cdot 4^{x+2}}[\/latex]<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"694071\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"694071\"]<\/p>\r\n<p>Simplify using laws of exponents:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} \\dfrac{(3x^{2})^3 \\cdot 2^{x-1}}{9x^{-1} \\cdot 4^{x+2}} &amp;= &amp; \\dfrac{3^3 \\cdot (x^2)^3 \\cdot 2^{x-1}}{3^2 \\cdot x^{-1} \\cdot (2^2)^{x+2}} \\\\ &amp;=&amp; \\dfrac{27x^6 \\cdot 2^{x-1}}{9x^{-1} \\cdot 2^{2(x+2)}} \\\\ &amp;=&amp; \\dfrac{27x^6 \\cdot 2^{x-1}}{9x^{-1} \\cdot 2^{2x+4}} \\\\ &amp;=&amp; \\dfrac{3x^7 \\cdot 2^{x-1}}{2^{2x+4}} \\\\ &amp;=&amp; 3x^7 \\cdot 2^{x-1-(2x+4)} \\\\ &amp;=&amp; 3x^7 \\cdot 2^{-x-5} \\end{array}[\/latex]<\/p>\r\n<p>Evaluate for [latex]x = 2[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]3(2^7) \\cdot 2^{-2-5} = 3 \\cdot 128 \\cdot 2^{-7} = 384 \\cdot \\frac{1}{128} = 3[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572452234\">Use the laws of exponents to simplify [latex]\\dfrac{(6x^{-3}y^2)}{(12x^{-4}y^5)}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"833456\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"833456\"]<\/p>\r\n<p id=\"fs-id1165042707513\">[latex]x^a\/x^b=x^{a-b}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572452533\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572452533\"]<\/p>\r\n<p>[latex]\\dfrac{x}{(2y^3)}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Logarithmic Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Definition of Logarithmic Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Inverse of exponential functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\log_b(x) = y[\/latex] if and only if [latex]b^y = x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Domain: [latex](0, \u221e)[\/latex], Range: [latex](-\u221e, \u221e)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Common Logarithms:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Base 10: [latex]\\log_{10}(x)[\/latex], often written as [latex]\\log(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Natural log: [latex]\\log_e(x)[\/latex], written as [latex]\\ln(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Properties of Logarithms:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Product: [latex]\\log_b(xy) = \\log_b(x) + \\log_b(y)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Quotient: [latex]\\log_b(\\frac{x}{y}) = \\log_b(x) - \\log_b(y)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Power: [latex]\\log_b(x^n) = n\\log_b(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Change-of-Base Formula:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\log_a(x) = \\dfrac{\\log_b(x)}{\\log_b(a)}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Useful for calculating logs with non-standard bases<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solving Logarithmic Equations:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Often involves converting to exponential form<\/li>\r\n\t<li class=\"whitespace-normal break-words\">May require using logarithm properties to simplify<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572232053\">Solve [latex]\\dfrac{e^{2x}}{(3+e^{2x})}=\\dfrac{1}{2}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"708533\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"708533\"]<\/p>\r\n<p id=\"fs-id1165042579095\">First solve the equation for [latex]e^{2x}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572174654\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572174654\"]<\/p>\r\n<p id=\"fs-id1170572174654\">[latex]x=\\dfrac{\\ln 3}{2}[\/latex]<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=827&amp;end=1126&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.5ExponentialAndLogarithmicFunctions827to1126_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.5 Exponential and Logarithmic Functions\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Solve [latex]\\ln(x^3)-4 \\ln (x)=1[\/latex].<\/p>\r\n<div id=\"fs-id1170572552646\" class=\"textbook key-takeaways\">\r\n<p>[reveal-answer q=\"662277\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"662277\"]<\/p>\r\n<p id=\"fs-id1165043161250\">First use the power property, then use the product property of logarithms.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572552698\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572552698\"]<\/p>\r\n<p id=\"fs-id1170572552698\">[latex]x=\\frac{1}{e}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572435068\">Use the change-of-base formula and a calculator to evaluate [latex]\\log_4 6[\/latex].<\/p>\r\n<p>[reveal-answer q=\"3762229\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3762229\"]<\/p>\r\n<p id=\"fs-id1165042853660\">Use the change of base to rewrite this expression in terms of expressions involving the natural logarithm function.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572128662\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572128662\"]<\/p>\r\n<p id=\"fs-id1170572128662\">[latex]1.29248[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Hyperbolic Functions\u00a0<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Definitions of Hyperbolic Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\cosh x = \\dfrac{e^x + e^{-x}}{2}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\sinh x = \\dfrac{e^x - e^{-x}}{2}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\tanh x = \\dfrac{\\sinh x}{\\cosh x} = \\frac{e^x - e^{-x}}{e^x + e^{-x}}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\text{csch} x = \\dfrac{1}{\\sinh x}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\text{sech} x = \\dfrac{1}{\\cosh x}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\coth x = \\dfrac{\\cosh x}{\\sinh x}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Graphs and Behavior:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\cosh x[\/latex]: Similar to [latex]|e^x|[\/latex], always [latex] \u2265 1[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\sinh x[\/latex]: Odd function, similar to [latex]e^x[\/latex] for large x<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\tanh x[\/latex]: Odd function, approaches [latex]\u00b11[\/latex] as [latex]x \u2192 \u00b1\u221e[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Identities:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\cosh^2 x - \\sinh^2 x = 1[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]1 - \\tanh^2 x = \\text{sech} ^2 x[\/latex]<\/li>\r\n\t<li>[latex]\\coth^2 x-1=\\text{csch}^2 x[\/latex]<\/li>\r\n\t<li>[latex]\\sinh(x \\pm y)=\\sinh x \\cosh y \\pm \\cosh x \\sinh y[\/latex]<\/li>\r\n\t<li>[latex]\\cosh (x \\pm y)=\\cosh x \\cosh y \\pm \\sinh x \\sinh y[\/latex]<\/li>\r\n\t<li>[latex]\\cosh x + \\sinh x = e^x[\/latex]<\/li>\r\n\t<li>[latex]\\cosh x-\\sinh x=e^{\u2212x}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Inverse Hyperbolic Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\sinh^{-1} x = \\ln(x + \\sqrt{x^2 + 1})[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\cosh^{-1} x = \\ln(x + \\sqrt{x^2 - 1})[\/latex] ([latex]x \u2265 1[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\tanh^{-1} x = \\frac{1}{2}\\ln(\\frac{1+x}{1-x})[\/latex] ([latex]|x| &lt; 1[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Simplify and evaluate [latex]\\sinh(\\cosh^{-1}(3))[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"820233\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"820233\"]<\/p>\r\n<p>Start with the inside function:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\cosh^{-1}(3) = \\ln(3 + \\sqrt{3^2 - 1}) = \\ln(3 + \\sqrt{8}) = \\ln(3 + 2\\sqrt{2})[\/latex]<\/p>\r\n<p>Now we have [latex]\\sinh(\\ln(3 + 2\\sqrt{2}))[\/latex]. Recall that [latex]\\sinh(\\ln x) = \\frac{x - \\frac{1}{x}}{2}[\/latex]<\/p>\r\n<p>Apply this:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\sinh(\\ln(3 + 2\\sqrt{2})) = \\dfrac{(3 + 2\\sqrt{2}) - \\dfrac{1}{3 + 2\\sqrt{2}}}{2}[\/latex]<\/p>\r\n<p>Simplify the denominator:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{1}{3 + 2\\sqrt{2}} = \\dfrac{3 - 2\\sqrt{2}}{(3 + 2\\sqrt{2})(3 - 2\\sqrt{2})} = \\dfrac{3 - 2\\sqrt{2}}{9 - 8} = 3 - 2\\sqrt{2}[\/latex]<\/p>\r\n<p>Substitute back:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{(3 + 2\\sqrt{2}) - (3 - 2\\sqrt{2})}{2} = \\dfrac{4\\sqrt{2}}{2} = 2\\sqrt{2}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572549916\">Simplify [latex]\\cosh(2 \\ln x)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"473309\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"473309\"]<\/p>\r\n<p id=\"fs-id1165039563336\">Use the definition of the cosh function and the power property of logarithm functions.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572549946\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572549946\"]<\/p>\r\n<p id=\"fs-id1170572549946\">[latex]\\frac{(x^2+x^{-2})}{2}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572176165\">Evaluate [latex]\\tanh^{-1}\\left(\\frac{1}{2}\\right)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"3088722\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3088722\"]<\/p>\r\n<p>Use the definition of [latex]\\tanh^{-1} x[\/latex] and simplify.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572176215\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572176215\"]<\/p>\r\n<p>[latex]\\frac{1}{2}\\ln(3) \\approx 0.5493[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify and evaluate exponential functions<\/li>\n<li>Analyze logarithmic functions by identifying forms, understanding their exponential relationships, and calculating different base logarithms<\/li>\n<li>Identify the hyperbolic functions, their graphs, and basic identities\u00a0<\/li>\n<\/ul>\n<\/section>\n<h2>Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Form: [latex]f(x) = ab^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is a non-zero real number (initial value)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex] is a positive real number, [latex]b \\neq 1[\/latex] (base)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Constant base, variable exponent<\/li>\n<li class=\"whitespace-normal break-words\">Rapid growth compared to power functions<\/li>\n<li class=\"whitespace-normal break-words\">Domain: all real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex](0, \u221e)[\/latex] for [latex]b > 1[\/latex], [latex](0, \u221e)[\/latex] for [latex]0 < b < 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluating Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute the given value for [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Follow order of operations<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to parentheses and exponents<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Laws of Exponents:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Product of Powers: [latex]b^x \\cdot b^y = b^{x+y}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Quotient of Powers: [latex]\\frac{b^x}{b^y} = b^{x-y}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Power of a Power: [latex](b^x)^y=b^{xy}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Power of a Product: [latex](ab)^x=a^x b^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Power of a Quotient: [latex]\\dfrac{a^x}{b^x} =\\left(\\dfrac{a}{b}\\right)^x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Population growth<\/li>\n<li class=\"whitespace-normal break-words\">Compound interest<\/li>\n<li class=\"whitespace-normal break-words\">Radioactive decay<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572173715\">Given the exponential function [latex]f(x)=100\u00b73^{x\/2}[\/latex], evaluate [latex]f(4)[\/latex] and [latex]f(10)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572173781\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572173781\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572173781\">[latex]f(4)=900; \\, f(10)=24,300[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox interact\">\n<p><a href=\"https:\/\/www.worldpopulationbalance.org\/understanding-exponential-growth\" target=\"_blank\" rel=\"noopener\">Go to World Population Balance for another example of exponential population growth.<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Simplify the following expression and then evaluate it for [latex]x = 2[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\dfrac{(3x^{2})^3 \\cdot 2^{x-1}}{9x^{-1} \\cdot 4^{x+2}}[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q694071\">Show Answer<\/button><\/p>\n<div id=\"q694071\" class=\"hidden-answer\" style=\"display: none\">\n<p>Simplify using laws of exponents:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} \\dfrac{(3x^{2})^3 \\cdot 2^{x-1}}{9x^{-1} \\cdot 4^{x+2}} &= & \\dfrac{3^3 \\cdot (x^2)^3 \\cdot 2^{x-1}}{3^2 \\cdot x^{-1} \\cdot (2^2)^{x+2}} \\\\ &=& \\dfrac{27x^6 \\cdot 2^{x-1}}{9x^{-1} \\cdot 2^{2(x+2)}} \\\\ &=& \\dfrac{27x^6 \\cdot 2^{x-1}}{9x^{-1} \\cdot 2^{2x+4}} \\\\ &=& \\dfrac{3x^7 \\cdot 2^{x-1}}{2^{2x+4}} \\\\ &=& 3x^7 \\cdot 2^{x-1-(2x+4)} \\\\ &=& 3x^7 \\cdot 2^{-x-5} \\end{array}[\/latex]<\/p>\n<p>Evaluate for [latex]x = 2[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]3(2^7) \\cdot 2^{-2-5} = 3 \\cdot 128 \\cdot 2^{-7} = 384 \\cdot \\frac{1}{128} = 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572452234\">Use the laws of exponents to simplify [latex]\\dfrac{(6x^{-3}y^2)}{(12x^{-4}y^5)}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q833456\">Hint<\/button><\/p>\n<div id=\"q833456\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042707513\">[latex]x^a\/x^b=x^{a-b}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572452533\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572452533\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{x}{(2y^3)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Logarithmic Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Logarithmic Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Inverse of exponential functions<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\log_b(x) = y[\/latex] if and only if [latex]b^y = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Domain: [latex](0, \u221e)[\/latex], Range: [latex](-\u221e, \u221e)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Common Logarithms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Base 10: [latex]\\log_{10}(x)[\/latex], often written as [latex]\\log(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Natural log: [latex]\\log_e(x)[\/latex], written as [latex]\\ln(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Properties of Logarithms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Product: [latex]\\log_b(xy) = \\log_b(x) + \\log_b(y)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Quotient: [latex]\\log_b(\\frac{x}{y}) = \\log_b(x) - \\log_b(y)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Power: [latex]\\log_b(x^n) = n\\log_b(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Change-of-Base Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_a(x) = \\dfrac{\\log_b(x)}{\\log_b(a)}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Useful for calculating logs with non-standard bases<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Logarithmic Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Often involves converting to exponential form<\/li>\n<li class=\"whitespace-normal break-words\">May require using logarithm properties to simplify<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572232053\">Solve [latex]\\dfrac{e^{2x}}{(3+e^{2x})}=\\dfrac{1}{2}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q708533\">Hint<\/button><\/p>\n<div id=\"q708533\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042579095\">First solve the equation for [latex]e^{2x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572174654\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572174654\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572174654\">[latex]x=\\dfrac{\\ln 3}{2}[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=827&amp;end=1126&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.5ExponentialAndLogarithmicFunctions827to1126_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.5 Exponential and Logarithmic Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Solve [latex]\\ln(x^3)-4 \\ln (x)=1[\/latex].<\/p>\n<div id=\"fs-id1170572552646\" class=\"textbook key-takeaways\">\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q662277\">Hint<\/button><\/p>\n<div id=\"q662277\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043161250\">First use the power property, then use the product property of logarithms.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572552698\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572552698\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572552698\">[latex]x=\\frac{1}{e}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572435068\">Use the change-of-base formula and a calculator to evaluate [latex]\\log_4 6[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3762229\">Hint<\/button><\/p>\n<div id=\"q3762229\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042853660\">Use the change of base to rewrite this expression in terms of expressions involving the natural logarithm function.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572128662\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572128662\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572128662\">[latex]1.29248[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Hyperbolic Functions\u00a0<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definitions of Hyperbolic Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\cosh x = \\dfrac{e^x + e^{-x}}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\sinh x = \\dfrac{e^x - e^{-x}}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tanh x = \\dfrac{\\sinh x}{\\cosh x} = \\frac{e^x - e^{-x}}{e^x + e^{-x}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\text{csch} x = \\dfrac{1}{\\sinh x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\text{sech} x = \\dfrac{1}{\\cosh x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\coth x = \\dfrac{\\cosh x}{\\sinh x}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphs and Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\cosh x[\/latex]: Similar to [latex]|e^x|[\/latex], always [latex]\u2265 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\sinh x[\/latex]: Odd function, similar to [latex]e^x[\/latex] for large x<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tanh x[\/latex]: Odd function, approaches [latex]\u00b11[\/latex] as [latex]x \u2192 \u00b1\u221e[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Identities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\cosh^2 x - \\sinh^2 x = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1 - \\tanh^2 x = \\text{sech} ^2 x[\/latex]<\/li>\n<li>[latex]\\coth^2 x-1=\\text{csch}^2 x[\/latex]<\/li>\n<li>[latex]\\sinh(x \\pm y)=\\sinh x \\cosh y \\pm \\cosh x \\sinh y[\/latex]<\/li>\n<li>[latex]\\cosh (x \\pm y)=\\cosh x \\cosh y \\pm \\sinh x \\sinh y[\/latex]<\/li>\n<li>[latex]\\cosh x + \\sinh x = e^x[\/latex]<\/li>\n<li>[latex]\\cosh x-\\sinh x=e^{\u2212x}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Hyperbolic Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\sinh^{-1} x = \\ln(x + \\sqrt{x^2 + 1})[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\cosh^{-1} x = \\ln(x + \\sqrt{x^2 - 1})[\/latex] ([latex]x \u2265 1[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tanh^{-1} x = \\frac{1}{2}\\ln(\\frac{1+x}{1-x})[\/latex] ([latex]|x| < 1[\/latex])<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Simplify and evaluate [latex]\\sinh(\\cosh^{-1}(3))[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q820233\">Show Answer<\/button><\/p>\n<div id=\"q820233\" class=\"hidden-answer\" style=\"display: none\">\n<p>Start with the inside function:<\/p>\n<p style=\"text-align: center;\">[latex]\\cosh^{-1}(3) = \\ln(3 + \\sqrt{3^2 - 1}) = \\ln(3 + \\sqrt{8}) = \\ln(3 + 2\\sqrt{2})[\/latex]<\/p>\n<p>Now we have [latex]\\sinh(\\ln(3 + 2\\sqrt{2}))[\/latex]. Recall that [latex]\\sinh(\\ln x) = \\frac{x - \\frac{1}{x}}{2}[\/latex]<\/p>\n<p>Apply this:<\/p>\n<p style=\"text-align: center;\">[latex]\\sinh(\\ln(3 + 2\\sqrt{2})) = \\dfrac{(3 + 2\\sqrt{2}) - \\dfrac{1}{3 + 2\\sqrt{2}}}{2}[\/latex]<\/p>\n<p>Simplify the denominator:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{1}{3 + 2\\sqrt{2}} = \\dfrac{3 - 2\\sqrt{2}}{(3 + 2\\sqrt{2})(3 - 2\\sqrt{2})} = \\dfrac{3 - 2\\sqrt{2}}{9 - 8} = 3 - 2\\sqrt{2}[\/latex]<\/p>\n<p>Substitute back:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{(3 + 2\\sqrt{2}) - (3 - 2\\sqrt{2})}{2} = \\dfrac{4\\sqrt{2}}{2} = 2\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572549916\">Simplify [latex]\\cosh(2 \\ln x)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q473309\">Hint<\/button><\/p>\n<div id=\"q473309\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165039563336\">Use the definition of the cosh function and the power property of logarithm functions.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572549946\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572549946\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572549946\">[latex]\\frac{(x^2+x^{-2})}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572176165\">Evaluate [latex]\\tanh^{-1}\\left(\\frac{1}{2}\\right)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3088722\">Hint<\/button><\/p>\n<div id=\"q3088722\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the definition of [latex]\\tanh^{-1} x[\/latex] and simplify.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572176215\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572176215\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{2}\\ln(3) \\approx 0.5493[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":759,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1179"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":20,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1179\/revisions"}],"predecessor-version":[{"id":3668,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1179\/revisions\/3668"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1179\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1179"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1179"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1179"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}