{"id":2288,"date":"2024-07-17T22:03:48","date_gmt":"2024-07-17T22:03:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2288"},"modified":"2025-08-15T15:27:22","modified_gmt":"2025-08-15T15:27:22","slug":"exponential-and-logarithmic-models-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponential-and-logarithmic-models-apply-it-1\/","title":{"raw":"Exponential and Logarithmic Models: Apply It 1","rendered":"Exponential and Logarithmic Models: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve real-world problems involving exponential and logarithmic equations.<\/li>\r\n \t<li>Create models for exponential growth and decay, including how to use Newton's Law of Cooling and logistic growth.<\/li>\r\n \t<li>Evaluate data to determine the most suitable model, with an emphasis on exponential contexts.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Building a Logarithmic Model from Data<\/h2>\r\nJust as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>logarithmic regression<\/h3>\r\nLogarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.\r\n\r\n&nbsp;\r\n\r\nThe logarithmic equation has the form\r\n<p style=\"text-align: center;\">[latex]y = a+b \\ln(x)[\/latex]<\/p>\r\nwhere\r\n<ul>\r\n \t<li>all input values, [latex]x[\/latex], must be non-negative.<\/li>\r\n \t<li>when [latex]b \\gt 0[\/latex], the model is increasing.<\/li>\r\n \t<li>when [latex]b \\lt 0[\/latex], the model is decreasing.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox interact\" aria-label=\"Interact\">Given a set of data, perform logarithmic regression using a graphing utility.\r\n<ol>\r\n \t<li>Use the STAT then EDIT menu to enter given data.\r\n<ol>\r\n \t<li>Clear any existing data from the lists.<\/li>\r\n \t<li>List the input values in the L1 column.<\/li>\r\n \t<li>List the output values in the L2 column.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\r\n<ol>\r\n \t<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\r\n \t<li>Verify the data follow a logarithmic pattern.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Find the equation that models the data.\r\n<ol>\r\n \t<li>Select \u201cLnReg\u201d from the STAT then CALC menu.<\/li>\r\n \t<li>Use the values returned for [latex]a[\/latex] and [latex]b[\/latex] to record the model.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">countries since the beginning of the 20th century.<\/span>Table below <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">shows the average life expectancies, in years, of Americans from 1900\u20132010.<\/span>\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">Year<\/td>\r\n<td style=\"width: 14.2857%;\">1900<\/td>\r\n<td style=\"width: 14.2857%;\">1910<\/td>\r\n<td style=\"width: 14.2857%;\">1920<\/td>\r\n<td style=\"width: 14.2857%;\">1930<\/td>\r\n<td style=\"width: 14.2857%;\">1940<\/td>\r\n<td style=\"width: 14.2857%;\">1950<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">Life Expectancy(Years)<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]47.3[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]50[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]54.1[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]59.7[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]62.9[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]68.2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">Year<\/td>\r\n<td style=\"width: 14.2857%;\">1960<\/td>\r\n<td style=\"width: 14.2857%;\">1970<\/td>\r\n<td style=\"width: 14.2857%;\">1980<\/td>\r\n<td style=\"width: 14.2857%;\">1990<\/td>\r\n<td style=\"width: 14.2857%;\">2000<\/td>\r\n<td style=\"width: 14.2857%;\">2010<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">Life Expectancy(Years<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]69.7[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]70.8[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]73.7[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]75.4[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]76.8[\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">[latex]78.7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Let [latex]x[\/latex] represent time in decades starting with [latex]x = 1[\/latex] for the year [latex]1900[\/latex], [latex]x= 2[\/latex] for the year [latex]1910[\/latex], etc. Let [latex]y[\/latex] represent the corresponding life expectancy. Use logarithmic regression to fit a model for this data.\r\n[reveal-answer q=\"178487\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"178487\"]Use the \u201cLnReg\u201d command from the STAT then CALC menu to obtain the logarithmic model, [latex]y = 42.52722583+13.85752327 \\ln(x)[\/latex].\r\nNext, graph the model in the same window as the scatterplot to verify it is a good fit\r\n\r\n[caption id=\"attachment_2289\" align=\"aligncenter\" width=\"351\"]<img class=\"wp-image-2289\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM.png\" alt=\"\" width=\"351\" height=\"239\" \/> Scatterplot with line of estimation[\/caption]\r\n\r\n[\/hidden-answer]<\/li>\r\n \t<li>Use the model to predict the average American life expectancy for the year [latex]2030[\/latex].\r\n[reveal-answer q=\"359573\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"359573\"]To predict the life expectancy of an American in the year 2030, substitute [latex]x = 14[\/latex] for the in the model and solve for [latex]y[\/latex]:\r\n[latex]y = 42.52722583+13.85752327 \\ln(x) = 42.52722583+13.85752327 \\ln(14) \\approx 79.1[\/latex].If life expectancy continues to increase at this pace, the average life expectancy of an American will be [latex]79.1[\/latex] by the year 2030.[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p class=\"whitespace-pre-wrap break-words\">As an environmental scientist, you're studying noise pollution in urban areas. Sound intensity is measured using decibels (dB), which follow a logarithmic scale. You're analyzing data to help city planners make informed decisions about noise reduction measures.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]31660[\/ohm2_question]<\/section>A city wants to compare two different methods of analyzing traffic noise. They collected the following data measuring average decibel levels during rush hour:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Time (hours past 6 AM)<\/th>\r\n<th>Decibel Level<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]72[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]78[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]82[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]84[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]85[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTwo models have been proposed:\r\n<ul>\r\n \t<li>Model A: [latex]y = 70 + 6.2ln(x)[\/latex]<\/li>\r\n \t<li>Model B: [latex]y = 73 + 5.1ln(x)[\/latex]<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]31661[\/ohm2_question]<\/section><section class=\"textbox connectIt\" aria-label=\"Connect It\">Support your answer by:\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>Calculating the predicted values at hours [latex]1[\/latex] and [latex]5[\/latex] using both models<\/li>\r\n \t<li>Finding which model has the smaller total error<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve real-world problems involving exponential and logarithmic equations.<\/li>\n<li>Create models for exponential growth and decay, including how to use Newton&#8217;s Law of Cooling and logistic growth.<\/li>\n<li>Evaluate data to determine the most suitable model, with an emphasis on exponential contexts.<\/li>\n<\/ul>\n<\/section>\n<h2>Building a Logarithmic Model from Data<\/h2>\n<p>Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>logarithmic regression<\/h3>\n<p>Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.<\/p>\n<p>&nbsp;<\/p>\n<p>The logarithmic equation has the form<\/p>\n<p style=\"text-align: center;\">[latex]y = a+b \\ln(x)[\/latex]<\/p>\n<p>where<\/p>\n<ul>\n<li>all input values, [latex]x[\/latex], must be non-negative.<\/li>\n<li>when [latex]b \\gt 0[\/latex], the model is increasing.<\/li>\n<li>when [latex]b \\lt 0[\/latex], the model is decreasing.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox interact\" aria-label=\"Interact\">Given a set of data, perform logarithmic regression using a graphing utility.<\/p>\n<ol>\n<li>Use the STAT then EDIT menu to enter given data.\n<ol>\n<li>Clear any existing data from the lists.<\/li>\n<li>List the input values in the L1 column.<\/li>\n<li>List the output values in the L2 column.<\/li>\n<\/ol>\n<\/li>\n<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\n<ol>\n<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\n<li>Verify the data follow a logarithmic pattern.<\/li>\n<\/ol>\n<\/li>\n<li>Find the equation that models the data.\n<ol>\n<li>Select \u201cLnReg\u201d from the STAT then CALC menu.<\/li>\n<li>Use the values returned for [latex]a[\/latex] and [latex]b[\/latex] to record the model.<\/li>\n<\/ol>\n<\/li>\n<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">countries since the beginning of the 20th century.<\/span>Table below <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">shows the average life expectancies, in years, of Americans from 1900\u20132010.<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 14.2857%;\">Year<\/td>\n<td style=\"width: 14.2857%;\">1900<\/td>\n<td style=\"width: 14.2857%;\">1910<\/td>\n<td style=\"width: 14.2857%;\">1920<\/td>\n<td style=\"width: 14.2857%;\">1930<\/td>\n<td style=\"width: 14.2857%;\">1940<\/td>\n<td style=\"width: 14.2857%;\">1950<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;\">Life Expectancy(Years)<\/td>\n<td style=\"width: 14.2857%;\">[latex]47.3[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]50[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]54.1[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]59.7[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]62.9[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]68.2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;\">Year<\/td>\n<td style=\"width: 14.2857%;\">1960<\/td>\n<td style=\"width: 14.2857%;\">1970<\/td>\n<td style=\"width: 14.2857%;\">1980<\/td>\n<td style=\"width: 14.2857%;\">1990<\/td>\n<td style=\"width: 14.2857%;\">2000<\/td>\n<td style=\"width: 14.2857%;\">2010<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;\">Life Expectancy(Years<\/td>\n<td style=\"width: 14.2857%;\">[latex]69.7[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]70.8[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]73.7[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]75.4[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]76.8[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">[latex]78.7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Let [latex]x[\/latex] represent time in decades starting with [latex]x = 1[\/latex] for the year [latex]1900[\/latex], [latex]x= 2[\/latex] for the year [latex]1910[\/latex], etc. Let [latex]y[\/latex] represent the corresponding life expectancy. Use logarithmic regression to fit a model for this data.\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q178487\">Show Answer<\/button><\/p>\n<div id=\"q178487\" class=\"hidden-answer\" style=\"display: none\">Use the \u201cLnReg\u201d command from the STAT then CALC menu to obtain the logarithmic model, [latex]y = 42.52722583+13.85752327 \\ln(x)[\/latex].<br \/>\nNext, graph the model in the same window as the scatterplot to verify it is a good fit<\/p>\n<figure id=\"attachment_2289\" aria-describedby=\"caption-attachment-2289\" style=\"width: 351px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2289\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM.png\" alt=\"\" width=\"351\" height=\"239\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM.png 1128w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM-300x204.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM-1024x697.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM-768x523.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM-65x44.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM-225x153.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/17220136\/Screenshot-2024-07-17-at-3.01.31%E2%80%AFPM-350x238.png 350w\" sizes=\"(max-width: 351px) 100vw, 351px\" \/><figcaption id=\"caption-attachment-2289\" class=\"wp-caption-text\">Scatterplot with line of estimation<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/li>\n<li>Use the model to predict the average American life expectancy for the year [latex]2030[\/latex].\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q359573\">Show Answer<\/button><\/p>\n<div id=\"q359573\" class=\"hidden-answer\" style=\"display: none\">To predict the life expectancy of an American in the year 2030, substitute [latex]x = 14[\/latex] for the in the model and solve for [latex]y[\/latex]:<br \/>\n[latex]y = 42.52722583+13.85752327 \\ln(x) = 42.52722583+13.85752327 \\ln(14) \\approx 79.1[\/latex].If life expectancy continues to increase at this pace, the average life expectancy of an American will be [latex]79.1[\/latex] by the year 2030.<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<p class=\"whitespace-pre-wrap break-words\">As an environmental scientist, you&#8217;re studying noise pollution in urban areas. Sound intensity is measured using decibels (dB), which follow a logarithmic scale. You&#8217;re analyzing data to help city planners make informed decisions about noise reduction measures.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm31660\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=31660&theme=lumen&iframe_resize_id=ohm31660&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>A city wants to compare two different methods of analyzing traffic noise. They collected the following data measuring average decibel levels during rush hour:<\/p>\n<table>\n<thead>\n<tr>\n<th>Time (hours past 6 AM)<\/th>\n<th>Decibel Level<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]72[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]78[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]82[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]84[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]85[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Two models have been proposed:<\/p>\n<ul>\n<li>Model A: [latex]y = 70 + 6.2ln(x)[\/latex]<\/li>\n<li>Model B: [latex]y = 73 + 5.1ln(x)[\/latex]<\/li>\n<\/ul>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm31661\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=31661&theme=lumen&iframe_resize_id=ohm31661&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Support your answer by:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Calculating the predicted values at hours [latex]1[\/latex] and [latex]5[\/latex] using both models<\/li>\n<li>Finding which model has the smaller total error<\/li>\n<\/ol>\n<\/section>\n","protected":false},"author":12,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":280,"module-header":"apply_it","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\/2288"}],"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\/12"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2288\/revisions"}],"predecessor-version":[{"id":4900,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2288\/revisions\/4900"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/280"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2288\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2288"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2288"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2288"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2288"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}