{"id":4811,"date":"2024-10-08T18:58:12","date_gmt":"2024-10-08T18:58:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=4811"},"modified":"2024-11-21T21:56:41","modified_gmt":"2024-11-21T21:56:41","slug":"exponential-and-logarithmic-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponential-and-logarithmic-functions-get-stronger\/","title":{"raw":"Exponential and Logarithmic Functions: Get Stronger","rendered":"Exponential and Logarithmic Functions: Get Stronger"},"content":{"raw":"<h2>Exponential Functions<\/h2>\r\nFor the following exercises, identify whether the statement represents an exponential function. Explain.\r\n<ol>\r\n \t<li>A population of bacteria decreases by a factor of [latex]\\dfrac{1}{8}[\/latex] every [latex]24[\/latex] hours.<\/li>\r\n \t<li>For each training session, a personal trainer charges his clients [latex]$5[\/latex] less than the previous training session.<\/li>\r\n<\/ol>\r\nFor the following exercises, consider this scenario: For each year [latex]t[\/latex], the population of a forest of trees is represented by the function [latex]A(t) = 115(1.025)^t[\/latex]. In a neighboring forest, the population of the same type of tree is represented by the function [latex]B(t) = 82(1.029)^t[\/latex]. (Round answers to the nearest whole number.)\r\n<ol start=\"3\">\r\n \t<li>Which forest\u2019s population is growing at a faster rate?<\/li>\r\n \t<li>Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after [latex]20[\/latex] years? By how many?<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.\r\n<ol start=\"5\">\r\n \t<li>[latex]y = 220(1.06)^x[\/latex]<\/li>\r\n \t<li>[latex]y = 11,701(0.97)^t[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the formula for an exponential function that passes through the two points given.\r\n<ol start=\"7\">\r\n \t<li>[latex](0, 2000)[\/latex] and [latex](2, 20)[\/latex]<\/li>\r\n \t<li>[latex]\\left(-1, \\dfrac{3}{2}\\right)[\/latex] and [latex](3, 24)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.\r\n<ol start=\"9\">\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<td>[latex]70[\/latex]<\/td>\r\n<td>[latex]40[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]-20[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]m(x)[\/latex]<\/th>\r\n<td>[latex]80[\/latex]<\/td>\r\n<td>[latex]61[\/latex]<\/td>\r\n<td>[latex]42.9[\/latex]<\/td>\r\n<td>[latex]25.61[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]g(x)[\/latex]<\/th>\r\n<td>[latex]-3.25[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]7.25[\/latex]<\/td>\r\n<td>[latex]12.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, evaluate each function. Round answers to four decimal places, if necessary.\r\n<ol start=\"12\">\r\n \t<li>[latex]f(x) = -4^{2x+3}[\/latex], for [latex]f(-1)[\/latex]<\/li>\r\n \t<li>[latex]f(x) = -2e^{x-1}[\/latex], for [latex]f(-1)[\/latex]<\/li>\r\n \t<li>[latex]f(x) = 1.2e^{2x} - 0.3[\/latex], for [latex]f(3)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine the equation of the new function, [latex]g(x)[\/latex]. State its [latex]y[\/latex]-intercept, domain, and range.\r\n<ol start=\"15\">\r\n \t<li>[latex]f(x) = 3^x[\/latex] is reflected about the [latex]y[\/latex]-axis and stretched vertically by a factor of [latex]4[\/latex]. What is the equation of the new function, [latex]g(x)[\/latex]?<\/li>\r\n \t<li>[latex]f(x) = 10^x[\/latex] is reflected about the [latex]x[\/latex]-axis and shifted upward [latex]7[\/latex] units. What is the equation of the new function, [latex]g(x)[\/latex]?<\/li>\r\n \t<li>[latex]f(x) = -\\dfrac{1}{2}\\left(\\dfrac{1}{4}\\right)^{x-2} + 4[\/latex] is shifted downward [latex]4[\/latex] units, then shifted left [latex]2[\/latex] units, stretched vertically by a factor of [latex]4[\/latex], and reflected about the [latex]x[\/latex]-axis. What is the equation of the new function, [latex]g(x)[\/latex]?<\/li>\r\n<\/ol>\r\nFor the following exercise, graph the function and its reflection about the [latex]y[\/latex]-axis on the same axes, and give the [latex]y[\/latex]-intercept.\r\n<ol start=\"18\">\r\n \t<li>[latex]g(x) = -2(0.25)^x[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, graph each set of functions on the same axes.\r\n<ol start=\"19\">\r\n \t<li>[latex]f(x) = 3\\left(\\dfrac{1}{4}\\right)^x[\/latex], [latex]g(x) = 3(2)^x[\/latex], and [latex]h(x) = 3(4)^x[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, match each function with one of the graphs in the figure below.\r\n<img class=\"alignnone size-medium wp-image-6132\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202615\/graphs_of_exponential_1_13-268x300.jpeg\" alt=\"\" width=\"268\" height=\"300\" \/>\r\n<ol start=\"20\">\r\n \t<li>[latex]f(x) = 2(0.69)^x[\/latex]<\/li>\r\n \t<li>[latex]f(x) = 2(0.81)^x[\/latex]<\/li>\r\n \t<li>[latex]f(x) = 2(1.59)^x[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graphs shown in the figure below. All have the form [latex]f(x) = ab^x[\/latex].\r\n<img class=\"alignnone size-medium wp-image-6134\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202724\/graphs_of_exponential_2_19-274x300.jpeg\" alt=\"\" width=\"274\" height=\"300\" \/>\r\n<ol start=\"23\">\r\n \t<li>Which graph has the largest value for [latex]b[\/latex]?<\/li>\r\n \t<li>Which graph has the largest value for [latex]a[\/latex]?<\/li>\r\n<\/ol>\r\nFor the following exercises, start with the graph of [latex]f(x) = 4^x[\/latex]. Then write a function that results from the given transformation.\r\n<ol start=\"25\">\r\n \t<li>Shift [latex]f(x)[\/latex] 3 units downward<\/li>\r\n \t<li>Shift [latex]f(x)[\/latex] 5 units right<\/li>\r\n \t<li>Reflect [latex]f(x)[\/latex] about the [latex]y[\/latex]-axis<\/li>\r\n<\/ol>\r\nFor the following exercise, each graph is a transformation of [latex]y=2^x[\/latex]. Write an equation describing the transformation.\r\n<ol start=\"28\">\r\n \t<li><img class=\"alignnone size-medium wp-image-6135\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203428\/graphs_of_exponential_3_39-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" \/><\/li>\r\n<\/ol>\r\nFor the following exercise, find an exponential equation for the graph.\r\n<ol start=\"29\">\r\n \t<li><img class=\"alignnone size-medium wp-image-6136\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203429\/graphs_of_exponential_4_41-274x300.jpeg\" alt=\"\" width=\"274\" height=\"300\" \/><\/li>\r\n<\/ol>\r\n<h2>Applications of Exponential Functions<\/h2>\r\nFor the following exercises, use the compound interest formula, [latex]A(t) = P\\left(1 + \\dfrac{r}{n}\\right)^{nt}[\/latex]. After a certain number of years, the value of an investment account is represented by the equation [latex]A = 10,250(1 + \\dfrac{0.04}{12})^{120}[\/latex].\r\n<ol>\r\n \t<li>What was the initial deposit made to the account?<\/li>\r\n \t<li>An account is opened with an initial deposit of [latex]$6,500[\/latex] and earns [latex]3.6\\%[\/latex] interest compounded semi-annually. What will the account be worth in [latex]20[\/latex] years?<\/li>\r\n \t<li>Solve the compound interest formula for the principal, [latex]P[\/latex].<\/li>\r\n \t<li>How much more would the account in the previous two exercises be worth if it were earning interest for [latex]5[\/latex] more years?<\/li>\r\n \t<li>Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of [latex]$9,000[\/latex] and was worth [latex]$13,373.53[\/latex] after [latex]10[\/latex] years.<\/li>\r\n<\/ol>\r\nFor the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.\r\n<ol start=\"6\">\r\n \t<li>[latex]y = 3742(e)^{0.75t}[\/latex]<\/li>\r\n \t<li>[latex]y = 2.25(e)^{-2t}[\/latex]<\/li>\r\n<\/ol>\r\n<ol start=\"8\">\r\n \t<li>The fox population in a certain region has an annual growth rate of [latex]9\\%[\/latex] per year. In the year [latex]2012[\/latex], there were [latex]23,900[\/latex] fox counted in the area. What is the fox population predicted to be in the year [latex]2020[\/latex]?<\/li>\r\n \t<li>In the year [latex]1985[\/latex], a house was valued at [latex]$110,000[\/latex]. By the year [latex]2005[\/latex], the value had appreciated to [latex]$145,000[\/latex]. What was the annual growth rate between [latex]1985[\/latex] and [latex]2005[\/latex]? Assume that the value continued to grow by the same percentage. What was the value of the house in the year [latex]2010[\/latex]?<\/li>\r\n \t<li>Jaylen wants to save [latex]$54,000[\/latex] for a down payment on a home. How much will he need to invest in an account with [latex]8.2\\%[\/latex] APR, compounding daily, in order to reach his goal in [latex]5[\/latex] years?<\/li>\r\n \t<li>Alyssa opened a retirement account with [latex]7.25\\%[\/latex] APR in the year [latex]2000[\/latex]. Her initial deposit was [latex]$13,500[\/latex]. How much will the account be worth in [latex]2025[\/latex] if interest compounds monthly? How much more would she make if interest compounded continuously?<\/li>\r\n<\/ol>\r\n<h2>Logarithmic Functions<\/h2>\r\nFor the following exercises, rewrite each equation in exponential form.\r\n<ol>\r\n \t<li>[latex]\\log_a(b) = c[\/latex]<\/li>\r\n \t<li>[latex]\\log_x(64) = y[\/latex]<\/li>\r\n \t<li>[latex]\\log_{15}(a) = b[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, rewrite each equation in logarithmic form.\r\n<ol start=\"4\">\r\n \t<li>[latex]c^d = k[\/latex]<\/li>\r\n \t<li>[latex]19^x = y[\/latex]<\/li>\r\n \t<li>[latex]y^x = \\dfrac{39}{100}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, solve for [latex]x[\/latex] by converting the logarithmic equation to exponential form.\r\n<ol start=\"7\">\r\n \t<li>[latex]\\log_2(x) = -3[\/latex]<\/li>\r\n \t<li>[latex]\\log_3(x) = 3[\/latex]<\/li>\r\n \t<li>[latex]\\log_9(x) = \\dfrac{1}{2}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the definition of common and natural logarithms to simplify.\r\n<ol start=\"10\">\r\n \t<li>[latex]10^{\\log(32)}[\/latex]<\/li>\r\n \t<li>[latex]e^{\\ln(1.06)}[\/latex]<\/li>\r\n \t<li>[latex]e^{\\ln(10.125)} + 4[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, evaluate the base [latex]b[\/latex] logarithmic expression without using a calculator.\r\n<ol start=\"13\">\r\n \t<li>[latex]\\log_6(\\sqrt{6})[\/latex]<\/li>\r\n \t<li>[latex]6\\log_8(4)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, evaluate the common logarithmic expression without using a calculator.\r\n<ol start=\"15\">\r\n \t<li>[latex]\\log(0.001)[\/latex]<\/li>\r\n \t<li>[latex]2\\log(100^{-3})[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, evaluate the natural logarithmic expression without using a calculator.\r\n<ol start=\"17\">\r\n \t<li>[latex]\\ln(1)[\/latex]<\/li>\r\n \t<li>[latex]25\\ln\\left(e^{\\frac{2}{5}}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<h2>Logarithmic Function Graphs and Characteristics<\/h2>\r\nFor the following exercises, state the domain and range of the function.\r\n<ol>\r\n \t<li>[latex]h(x) = \\ln\\left(\\dfrac{1}{2} - x\\right)[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\ln(4x + 17) - 5[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, state the domain and the vertical asymptote of the function.\r\n<ol start=\"3\">\r\n \t<li>[latex]f(x) = \\log_b(x - 5)[\/latex]<\/li>\r\n \t<li>[latex]f(x) = \\log(3x + 1)[\/latex]<\/li>\r\n \t<li>[latex]g(x) = -\\ln(3x + 9) - 7[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, state the domain, range, and [latex]x[\/latex]- and [latex]y[\/latex]-intercepts, if they exist. If they do not exist, write [latex]DNE[\/latex].\r\n<ol start=\"6\">\r\n \t<li>[latex]h(x) = \\log_4(x - 1) + 1[\/latex]<\/li>\r\n \t<li>[latex]g(x) = \\ln(-x) - 2[\/latex]<\/li>\r\n \t<li>[latex]h(x) = 3\\ln(x) - 9[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, match each function in the figure below with the letter corresponding to its graph.\r\n<img class=\"alignnone size-medium wp-image-6139\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212506\/graphs_of_logarithmic_1_27-300x271.jpeg\" alt=\"\" width=\"300\" height=\"271\" \/>\r\n<ol start=\"9\">\r\n \t<li>[latex]f(x) = \\ln(x)[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\log_5(x)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, match each function in the figure below with the letter corresponding to its graph.\r\n<img class=\"alignnone size-medium wp-image-6140\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212542\/graphs_of_logarithmic_2_31-233x300.jpeg\" alt=\"\" width=\"233\" height=\"300\" \/>\r\n<ol start=\"11\">\r\n \t<li>[latex]f(x) = \\log_{\\frac{1}{3}}(x)[\/latex]<\/li>\r\n \t<li>[latex]h(x) = \\log_{\\frac{3}{4}}(x)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, sketch the graphs of each pair of functions on the same axis.\r\n<ol start=\"13\">\r\n \t<li>[latex]f(x) = \\log(x)[\/latex] and [latex]g(x) = \\log_{\\frac{1}{2}}(x)[\/latex]<\/li>\r\n \t<li>[latex]f(x) = e^x[\/latex] and [latex]g(x) = \\ln(x)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, sketch the graph of the indicated function.\r\n<ol start=\"15\">\r\n \t<li>[latex]f(x) = \\log_2(x + 2)[\/latex]<\/li>\r\n \t<li>[latex]f(x) = \\ln(-x)[\/latex]<\/li>\r\n \t<li>[latex]g(x) = \\log(6 - 3x) + 1[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write a logarithmic equation corresponding to the graph shown.\r\n<ol start=\"18\">\r\n \t<li>Use [latex]y = \\log_2(x)[\/latex] as the parent function.\r\n<img class=\"alignnone size-medium wp-image-6142\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213351\/graphs_of_logarithmic_4_47-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/><\/li>\r\n \t<li>Use [latex]f(x) = \\log_4(x)[\/latex] as the parent function.\r\n<img class=\"alignnone size-medium wp-image-6143\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213352\/graphs_of_logarithmic_5_49-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/><\/li>\r\n<\/ol>","rendered":"<h2>Exponential Functions<\/h2>\n<p>For the following exercises, identify whether the statement represents an exponential function. Explain.<\/p>\n<ol>\n<li>A population of bacteria decreases by a factor of [latex]\\dfrac{1}{8}[\/latex] every [latex]24[\/latex] hours.<\/li>\n<li>For each training session, a personal trainer charges his clients [latex]$5[\/latex] less than the previous training session.<\/li>\n<\/ol>\n<p>For the following exercises, consider this scenario: For each year [latex]t[\/latex], the population of a forest of trees is represented by the function [latex]A(t) = 115(1.025)^t[\/latex]. In a neighboring forest, the population of the same type of tree is represented by the function [latex]B(t) = 82(1.029)^t[\/latex]. (Round answers to the nearest whole number.)<\/p>\n<ol start=\"3\">\n<li>Which forest\u2019s population is growing at a faster rate?<\/li>\n<li>Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after [latex]20[\/latex] years? By how many?<\/li>\n<\/ol>\n<p>For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.<\/p>\n<ol start=\"5\">\n<li>[latex]y = 220(1.06)^x[\/latex]<\/li>\n<li>[latex]y = 11,701(0.97)^t[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, find the formula for an exponential function that passes through the two points given.<\/p>\n<ol start=\"7\">\n<li>[latex](0, 2000)[\/latex] and [latex](2, 20)[\/latex]<\/li>\n<li>[latex]\\left(-1, \\dfrac{3}{2}\\right)[\/latex] and [latex](3, 24)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.<\/p>\n<ol start=\"9\">\n<li>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<th>[latex]f(x)[\/latex]<\/th>\n<td>[latex]70[\/latex]<\/td>\n<td>[latex]40[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]-20[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<th>[latex]m(x)[\/latex]<\/th>\n<td>[latex]80[\/latex]<\/td>\n<td>[latex]61[\/latex]<\/td>\n<td>[latex]42.9[\/latex]<\/td>\n<td>[latex]25.61[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<th>[latex]g(x)[\/latex]<\/th>\n<td>[latex]-3.25[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]7.25[\/latex]<\/td>\n<td>[latex]12.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<p>For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.<\/p>\n<ol start=\"12\">\n<li>[latex]f(x) = -4^{2x+3}[\/latex], for [latex]f(-1)[\/latex]<\/li>\n<li>[latex]f(x) = -2e^{x-1}[\/latex], for [latex]f(-1)[\/latex]<\/li>\n<li>[latex]f(x) = 1.2e^{2x} - 0.3[\/latex], for [latex]f(3)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, determine the equation of the new function, [latex]g(x)[\/latex]. State its [latex]y[\/latex]-intercept, domain, and range.<\/p>\n<ol start=\"15\">\n<li>[latex]f(x) = 3^x[\/latex] is reflected about the [latex]y[\/latex]-axis and stretched vertically by a factor of [latex]4[\/latex]. What is the equation of the new function, [latex]g(x)[\/latex]?<\/li>\n<li>[latex]f(x) = 10^x[\/latex] is reflected about the [latex]x[\/latex]-axis and shifted upward [latex]7[\/latex] units. What is the equation of the new function, [latex]g(x)[\/latex]?<\/li>\n<li>[latex]f(x) = -\\dfrac{1}{2}\\left(\\dfrac{1}{4}\\right)^{x-2} + 4[\/latex] is shifted downward [latex]4[\/latex] units, then shifted left [latex]2[\/latex] units, stretched vertically by a factor of [latex]4[\/latex], and reflected about the [latex]x[\/latex]-axis. What is the equation of the new function, [latex]g(x)[\/latex]?<\/li>\n<\/ol>\n<p>For the following exercise, graph the function and its reflection about the [latex]y[\/latex]-axis on the same axes, and give the [latex]y[\/latex]-intercept.<\/p>\n<ol start=\"18\">\n<li>[latex]g(x) = -2(0.25)^x[\/latex]<\/li>\n<\/ol>\n<p>For the following exercise, graph each set of functions on the same axes.<\/p>\n<ol start=\"19\">\n<li>[latex]f(x) = 3\\left(\\dfrac{1}{4}\\right)^x[\/latex], [latex]g(x) = 3(2)^x[\/latex], and [latex]h(x) = 3(4)^x[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, match each function with one of the graphs in the figure below.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6132\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202615\/graphs_of_exponential_1_13-268x300.jpeg\" alt=\"\" width=\"268\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202615\/graphs_of_exponential_1_13-268x300.jpeg 268w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202615\/graphs_of_exponential_1_13-65x73.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202615\/graphs_of_exponential_1_13-225x252.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202615\/graphs_of_exponential_1_13.jpeg 347w\" sizes=\"(max-width: 268px) 100vw, 268px\" \/><\/p>\n<ol start=\"20\">\n<li>[latex]f(x) = 2(0.69)^x[\/latex]<\/li>\n<li>[latex]f(x) = 2(0.81)^x[\/latex]<\/li>\n<li>[latex]f(x) = 2(1.59)^x[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the graphs shown in the figure below. All have the form [latex]f(x) = ab^x[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6134\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202724\/graphs_of_exponential_2_19-274x300.jpeg\" alt=\"\" width=\"274\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202724\/graphs_of_exponential_2_19-274x300.jpeg 274w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202724\/graphs_of_exponential_2_19-65x71.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202724\/graphs_of_exponential_2_19-225x246.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202724\/graphs_of_exponential_2_19-350x383.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31202724\/graphs_of_exponential_2_19.jpeg 383w\" sizes=\"(max-width: 274px) 100vw, 274px\" \/><\/p>\n<ol start=\"23\">\n<li>Which graph has the largest value for [latex]b[\/latex]?<\/li>\n<li>Which graph has the largest value for [latex]a[\/latex]?<\/li>\n<\/ol>\n<p>For the following exercises, start with the graph of [latex]f(x) = 4^x[\/latex]. Then write a function that results from the given transformation.<\/p>\n<ol start=\"25\">\n<li>Shift [latex]f(x)[\/latex] 3 units downward<\/li>\n<li>Shift [latex]f(x)[\/latex] 5 units right<\/li>\n<li>Reflect [latex]f(x)[\/latex] about the [latex]y[\/latex]-axis<\/li>\n<\/ol>\n<p>For the following exercise, each graph is a transformation of [latex]y=2^x[\/latex]. Write an equation describing the transformation.<\/p>\n<ol start=\"28\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6135\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203428\/graphs_of_exponential_3_39-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203428\/graphs_of_exponential_3_39-290x300.jpeg 290w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203428\/graphs_of_exponential_3_39-65x67.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203428\/graphs_of_exponential_3_39-225x233.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203428\/graphs_of_exponential_3_39-350x362.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203428\/graphs_of_exponential_3_39.jpeg 360w\" sizes=\"(max-width: 290px) 100vw, 290px\" \/><\/li>\n<\/ol>\n<p>For the following exercise, find an exponential equation for the graph.<\/p>\n<ol start=\"29\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6136\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203429\/graphs_of_exponential_4_41-274x300.jpeg\" alt=\"\" width=\"274\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203429\/graphs_of_exponential_4_41-274x300.jpeg 274w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203429\/graphs_of_exponential_4_41-65x71.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203429\/graphs_of_exponential_4_41-225x246.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31203429\/graphs_of_exponential_4_41.jpeg 341w\" sizes=\"(max-width: 274px) 100vw, 274px\" \/><\/li>\n<\/ol>\n<h2>Applications of Exponential Functions<\/h2>\n<p>For the following exercises, use the compound interest formula, [latex]A(t) = P\\left(1 + \\dfrac{r}{n}\\right)^{nt}[\/latex]. After a certain number of years, the value of an investment account is represented by the equation [latex]A = 10,250(1 + \\dfrac{0.04}{12})^{120}[\/latex].<\/p>\n<ol>\n<li>What was the initial deposit made to the account?<\/li>\n<li>An account is opened with an initial deposit of [latex]$6,500[\/latex] and earns [latex]3.6\\%[\/latex] interest compounded semi-annually. What will the account be worth in [latex]20[\/latex] years?<\/li>\n<li>Solve the compound interest formula for the principal, [latex]P[\/latex].<\/li>\n<li>How much more would the account in the previous two exercises be worth if it were earning interest for [latex]5[\/latex] more years?<\/li>\n<li>Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of [latex]$9,000[\/latex] and was worth [latex]$13,373.53[\/latex] after [latex]10[\/latex] years.<\/li>\n<\/ol>\n<p>For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.<\/p>\n<ol start=\"6\">\n<li>[latex]y = 3742(e)^{0.75t}[\/latex]<\/li>\n<li>[latex]y = 2.25(e)^{-2t}[\/latex]<\/li>\n<\/ol>\n<ol start=\"8\">\n<li>The fox population in a certain region has an annual growth rate of [latex]9\\%[\/latex] per year. In the year [latex]2012[\/latex], there were [latex]23,900[\/latex] fox counted in the area. What is the fox population predicted to be in the year [latex]2020[\/latex]?<\/li>\n<li>In the year [latex]1985[\/latex], a house was valued at [latex]$110,000[\/latex]. By the year [latex]2005[\/latex], the value had appreciated to [latex]$145,000[\/latex]. What was the annual growth rate between [latex]1985[\/latex] and [latex]2005[\/latex]? Assume that the value continued to grow by the same percentage. What was the value of the house in the year [latex]2010[\/latex]?<\/li>\n<li>Jaylen wants to save [latex]$54,000[\/latex] for a down payment on a home. How much will he need to invest in an account with [latex]8.2\\%[\/latex] APR, compounding daily, in order to reach his goal in [latex]5[\/latex] years?<\/li>\n<li>Alyssa opened a retirement account with [latex]7.25\\%[\/latex] APR in the year [latex]2000[\/latex]. Her initial deposit was [latex]$13,500[\/latex]. How much will the account be worth in [latex]2025[\/latex] if interest compounds monthly? How much more would she make if interest compounded continuously?<\/li>\n<\/ol>\n<h2>Logarithmic Functions<\/h2>\n<p>For the following exercises, rewrite each equation in exponential form.<\/p>\n<ol>\n<li>[latex]\\log_a(b) = c[\/latex]<\/li>\n<li>[latex]\\log_x(64) = y[\/latex]<\/li>\n<li>[latex]\\log_{15}(a) = b[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, rewrite each equation in logarithmic form.<\/p>\n<ol start=\"4\">\n<li>[latex]c^d = k[\/latex]<\/li>\n<li>[latex]19^x = y[\/latex]<\/li>\n<li>[latex]y^x = \\dfrac{39}{100}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, solve for [latex]x[\/latex] by converting the logarithmic equation to exponential form.<\/p>\n<ol start=\"7\">\n<li>[latex]\\log_2(x) = -3[\/latex]<\/li>\n<li>[latex]\\log_3(x) = 3[\/latex]<\/li>\n<li>[latex]\\log_9(x) = \\dfrac{1}{2}[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, use the definition of common and natural logarithms to simplify.<\/p>\n<ol start=\"10\">\n<li>[latex]10^{\\log(32)}[\/latex]<\/li>\n<li>[latex]e^{\\ln(1.06)}[\/latex]<\/li>\n<li>[latex]e^{\\ln(10.125)} + 4[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, evaluate the base [latex]b[\/latex] logarithmic expression without using a calculator.<\/p>\n<ol start=\"13\">\n<li>[latex]\\log_6(\\sqrt{6})[\/latex]<\/li>\n<li>[latex]6\\log_8(4)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, evaluate the common logarithmic expression without using a calculator.<\/p>\n<ol start=\"15\">\n<li>[latex]\\log(0.001)[\/latex]<\/li>\n<li>[latex]2\\log(100^{-3})[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, evaluate the natural logarithmic expression without using a calculator.<\/p>\n<ol start=\"17\">\n<li>[latex]\\ln(1)[\/latex]<\/li>\n<li>[latex]25\\ln\\left(e^{\\frac{2}{5}}\\right)[\/latex]<\/li>\n<\/ol>\n<h2>Logarithmic Function Graphs and Characteristics<\/h2>\n<p>For the following exercises, state the domain and range of the function.<\/p>\n<ol>\n<li>[latex]h(x) = \\ln\\left(\\dfrac{1}{2} - x\\right)[\/latex]<\/li>\n<li>[latex]h(x) = \\ln(4x + 17) - 5[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, state the domain and the vertical asymptote of the function.<\/p>\n<ol start=\"3\">\n<li>[latex]f(x) = \\log_b(x - 5)[\/latex]<\/li>\n<li>[latex]f(x) = \\log(3x + 1)[\/latex]<\/li>\n<li>[latex]g(x) = -\\ln(3x + 9) - 7[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, state the domain, range, and [latex]x[\/latex]&#8211; and [latex]y[\/latex]-intercepts, if they exist. If they do not exist, write [latex]DNE[\/latex].<\/p>\n<ol start=\"6\">\n<li>[latex]h(x) = \\log_4(x - 1) + 1[\/latex]<\/li>\n<li>[latex]g(x) = \\ln(-x) - 2[\/latex]<\/li>\n<li>[latex]h(x) = 3\\ln(x) - 9[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, match each function in the figure below with the letter corresponding to its graph.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6139\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212506\/graphs_of_logarithmic_1_27-300x271.jpeg\" alt=\"\" width=\"300\" height=\"271\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212506\/graphs_of_logarithmic_1_27-300x271.jpeg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212506\/graphs_of_logarithmic_1_27-65x59.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212506\/graphs_of_logarithmic_1_27-225x203.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212506\/graphs_of_logarithmic_1_27-350x316.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212506\/graphs_of_logarithmic_1_27.jpeg 487w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<ol start=\"9\">\n<li>[latex]f(x) = \\ln(x)[\/latex]<\/li>\n<li>[latex]h(x) = \\log_5(x)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, match each function in the figure below with the letter corresponding to its graph.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6140\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212542\/graphs_of_logarithmic_2_31-233x300.jpeg\" alt=\"\" width=\"233\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212542\/graphs_of_logarithmic_2_31-233x300.jpeg 233w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212542\/graphs_of_logarithmic_2_31-65x84.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212542\/graphs_of_logarithmic_2_31-225x289.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31212542\/graphs_of_logarithmic_2_31.jpeg 342w\" sizes=\"(max-width: 233px) 100vw, 233px\" \/><\/p>\n<ol start=\"11\">\n<li>[latex]f(x) = \\log_{\\frac{1}{3}}(x)[\/latex]<\/li>\n<li>[latex]h(x) = \\log_{\\frac{3}{4}}(x)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, sketch the graphs of each pair of functions on the same axis.<\/p>\n<ol start=\"13\">\n<li>[latex]f(x) = \\log(x)[\/latex] and [latex]g(x) = \\log_{\\frac{1}{2}}(x)[\/latex]<\/li>\n<li>[latex]f(x) = e^x[\/latex] and [latex]g(x) = \\ln(x)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, sketch the graph of the indicated function.<\/p>\n<ol start=\"15\">\n<li>[latex]f(x) = \\log_2(x + 2)[\/latex]<\/li>\n<li>[latex]f(x) = \\ln(-x)[\/latex]<\/li>\n<li>[latex]g(x) = \\log(6 - 3x) + 1[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write a logarithmic equation corresponding to the graph shown.<\/p>\n<ol start=\"18\">\n<li>Use [latex]y = \\log_2(x)[\/latex] as the parent function.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6142\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213351\/graphs_of_logarithmic_4_47-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213351\/graphs_of_logarithmic_4_47-298x300.jpeg 298w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213351\/graphs_of_logarithmic_4_47-150x150.jpeg 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213351\/graphs_of_logarithmic_4_47-65x66.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213351\/graphs_of_logarithmic_4_47-225x227.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213351\/graphs_of_logarithmic_4_47-350x353.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213351\/graphs_of_logarithmic_4_47.jpeg 375w\" sizes=\"(max-width: 298px) 100vw, 298px\" \/><\/li>\n<li>Use [latex]f(x) = \\log_4(x)[\/latex] as the parent function.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-6143\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213352\/graphs_of_logarithmic_5_49-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213352\/graphs_of_logarithmic_5_49-298x300.jpeg 298w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213352\/graphs_of_logarithmic_5_49-150x150.jpeg 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213352\/graphs_of_logarithmic_5_49-65x66.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213352\/graphs_of_logarithmic_5_49-225x227.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213352\/graphs_of_logarithmic_5_49-350x353.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/31213352\/graphs_of_logarithmic_5_49.jpeg 375w\" sizes=\"(max-width: 298px) 100vw, 298px\" \/><\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":31,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":255,"module-header":"practice","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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4811"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4811\/revisions"}],"predecessor-version":[{"id":6338,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4811\/revisions\/6338"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/255"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4811\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=4811"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4811"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=4811"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=4811"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}