{"id":1449,"date":"2025-07-25T01:30:14","date_gmt":"2025-07-25T01:30:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1449"},"modified":"2026-03-11T09:13:51","modified_gmt":"2026-03-11T09:13:51","slug":"fitting-exponential-models-to-data-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/fitting-exponential-models-to-data-fresh-take\/","title":{"raw":"Fitting Exponential Models to Data: Fresh Take","rendered":"Fitting Exponential Models to Data: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Build an exponential model from data<\/li>\r\n \t<li>Build a logarithmic model from data<\/li>\r\n \t<li>Build a logistic model from data<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Build an Exponential Model from Data<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Exponential Regression:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used when data grows or decays at an increasing rate<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Model form: [latex]y = ab^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]b &gt; 1[\/latex]: exponential growth<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]0 &lt; b &lt; 1[\/latex]: exponential decay<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Using a Graphing Calculator:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Enter data in lists L1 (x-values) and L2 (y-values)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Create a scatter plot to verify exponential pattern<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use \"ExpReg\" from STAT CALC menu<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Record values for [latex]a[\/latex] and [latex]b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using Technology (Desmos):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Create a table with your data<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Enter [latex]y_1 \\sim ab^{x_1}[\/latex] below the table<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The tool displays values for [latex]a[\/latex] and [latex]b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">A credit card balance grows each month after graduation. The table shows the debt over 8 months:[latex]\\begin{array}{|c|c|c|c|c|c|c|c|c|}\r\n\\hline\r\n\\text{Month} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 \\\\\r\n\\hline\r\n\\text{Debt (\\$)} &amp; 620 &amp; 761.88 &amp; 899.80 &amp; 1039.93 &amp; 1270.63 &amp; 1589.04 &amp; 1851.31 &amp; 2154.92 \\\\\r\n\\hline\r\n\\end{array}[\/latex]Use exponential regression to fit a model to these data. If spending continues at this rate, what will the debt be after one year (month 12)?[reveal-answer q=\"exp-reg-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"exp-reg-001\"]\r\nEnter the data into a graphing utility and use exponential regression.The model is: [latex]y = 522.89(1.196)^x[\/latex]To find the debt after 12 months:\r\n[latex]\\begin{align*}\r\ny &amp;= 522.89(1.196)^x \\\\\r\ny &amp;= 522.89(1.196)^{12} \\\\\r\ny &amp;\\approx 4499.38\r\n\\end{align*}[\/latex]The credit card debt will be approximately $4,499.38 after one year.\r\n[\/hidden-answer]<\/section><section class=\"textbox youChoose\" aria-label=\"You Choose\">[videopicker divId=\"tnh-video-picker\" title=\"Choose a Calculator\" label=\"Select Calculator Type\"][videooption displayName=\"TI-84 Plus CE\" value=\"\/\/plugin.3playmedia.com\/show?mf=14660359&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=WpBmF3Ma0Hs&amp;video_target=tpm-plugin-drvnscs7-WpBmF3Ma0Hs\"] [videooption displayName=\"Desmos\" value=\"\/\/plugin.3playmedia.com\/show?mf=14660360&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=O-hx7i3TkEg&amp;video_target=tpm-plugin-drvnscs7-O-hx7i3TkEg\"][\/videopicker]You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+do+Exponential+Regression+on+the+TI84+Plus+CE_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to do Exponential Regression on the TI84 Plus CE\u201d here (opens in new window).<\/a>\r\nYou can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Exponential+Regression+-+Desmos_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential Regression - Desmos\u201d here (opens in new window).<\/a><\/section>\r\n<h2>Build a Logarithmic Model from Data<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Logarithmic Regression:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used when data increases or decreases rapidly at first, then slows<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Model form: [latex]y = a + b\\ln(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">All input values must be positive<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]b &gt; 0[\/latex]: increasing model<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]b &lt; 0[\/latex]: decreasing model<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Using a Graphing Calculator:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Enter data in lists L1 and L2<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify scatter plot shows logarithmic pattern<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use \"LnReg\" from STAT CALC menu<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Record model as [latex]y = a + b\\ln(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Life expectancy has been increasing since 1900. The table shows average life expectancies of Americans:[latex]\\begin{array}{|c|c|c|c|c|c|c|}\r\n\\hline\r\n\\text{Year} &amp; 1900 &amp; 1910 &amp; 1920 &amp; 1930 &amp; 1940 &amp; 1950 \\\\\r\n\\hline\r\n\\text{Life Exp.} &amp; 47.3 &amp; 50.0 &amp; 54.1 &amp; 59.7 &amp; 62.9 &amp; 68.2 \\\\\r\n\\hline\r\n\\end{array}[\/latex][latex]\\begin{array}{|c|c|c|c|c|c|c|}\r\n\\hline\r\n\\text{Year} &amp; 1960 &amp; 1970 &amp; 1980 &amp; 1990 &amp; 2000 &amp; 2010 \\\\\r\n\\hline\r\n\\text{Life Exp.} &amp; 69.7 &amp; 70.8 &amp; 73.7 &amp; 75.4 &amp; 76.8 &amp; 78.7 \\\\\r\n\\hline\r\n\\end{array}[\/latex]Let [latex]x = 1[\/latex] for 1900, [latex]x = 2[\/latex] for 1910, etc. Use logarithmic regression to predict life expectancy in 2030.[reveal-answer q=\"log-reg-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"log-reg-001\"]\r\nEnter the data using [latex]x[\/latex] values 1\u201312 and use logarithmic regression.The model is: [latex]y = 42.53 + 13.86\\ln(x)[\/latex]For 2030, use [latex]x = 14[\/latex]:\r\n[latex]\\begin{align*}\r\ny &amp;= 42.53 + 13.86\\ln(14) \\\\\r\ny &amp;\\approx 79.1\r\n\\end{align*}[\/latex]The predicted life expectancy for 2030 is approximately 79.1 years.\r\n[\/hidden-answer]<\/section><section class=\"textbox watchIt\">[videopicker divId=\"tnh-video-picker\" title=\"Choose a Calculator\" label=\"Select Calculator\"][videooption displayName=\"TI-84+\" value=\"\/\/plugin.3playmedia.com\/show?mf=14660361&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=YCRgsUSotEY&amp;video_target=tpm-plugin-drvnscs7-YCRgsUSotEY\"]\r\n[videooption displayName=\"Desmos\" value=\"\/\/plugin.3playmedia.com\/show?mf=14660362&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pqM1hzHMx9g&amp;video_target=tpm-plugin-drvnscs7-pqM1hzHMx9g\"]\r\n[\/videopicker]You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Logarithmic+Regression+on+the+TI84_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithmic Regression on the TI84\u201d here (opens in new window).<\/a>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Desmos+Logarithmic+Regression_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDesmos Logarithmic Regression\u201d here (opens in new window).<\/a><\/section>\r\n<h2>Build a Logistic Model from Data<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Logistic Regression:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used when growth starts rapidly then levels off to an upper limit<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Model form: [latex]y = \\frac{c}{1 + ae^{-bx}}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]c[\/latex] is the limiting value (carrying capacity)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Initial value is [latex]\\frac{c}{1+a}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Using a Graphing Calculator:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Enter data in lists L1 and L2<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify scatter plot shows logistic pattern (S-shaped curve)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use \"Logistic\" from STAT CALC menu<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Record values for [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Harbor seal populations in the Wadden Sea from 1997-2012:[latex]\\begin{array}{|c|c|c|c|c|}\r\n\\hline\r\n\\text{Year} &amp; 1997 &amp; 1998 &amp; 1999 &amp; 2000 \\\\\r\n\\hline\r\n\\text{Population (thousands)} &amp; 3.493 &amp; 5.282 &amp; 6.357 &amp; 9.201 \\\\\r\n\\hline\r\n\\end{array}[\/latex](data continues through 2012 with population reaching 25.108)Let [latex]x = 0[\/latex] for 1997. Use logistic regression to:\r\na. Fit a model to the data\r\nb. Predict seal population in 2020\r\nc. Find the carrying capacity[reveal-answer q=\"log-reg-002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"log-reg-002\"]\r\nUsing logistic regression:a. The model is: [latex]y = \\frac{25.66}{1 + 6.11e^{-0.385x}}[\/latex]b. For 2020, use [latex]x = 23[\/latex]:\r\n[latex]y = \\frac{25.66}{1 + 6.11e^{-0.385(23)}} \\approx 25.63[\/latex]About 25,634 seals in 2020.c. The carrying capacity is [latex]c \\approx 25,657[\/latex] seals.\r\n[\/hidden-answer]<\/section><section class=\"textbox watchIt\">[videopicker divId=\"tnh-video-picker\" title=\"Choose a Calculator\" label=\"Select Calculator\"][videooption displayName=\"TI-84+\" value=\"\/\/plugin.3playmedia.com\/show?mf=14660363&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fRiTk2ywiio&amp;video_target=tpm-plugin-drvnscs7-fRiTk2ywiio\"]\r\n[videooption displayName=\"Desmos\" value=\"\/\/plugin.3playmedia.com\/show?mf=14679710&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=b3LsIcZzrHI&amp;video_target=tpm-plugin-drvnscs7-b3LsIcZzrHI\"]\r\n[\/videopicker]You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Logistic+Regression+on+the+TI84_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogistic Regression on the TI84\u201d here (opens in new window).<\/a>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/5+7+Logistical+Regression_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c5 7 Logistical Regression\u201d here (opens in new window).<\/a><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Build an exponential model from data<\/li>\n<li>Build a logarithmic model from data<\/li>\n<li>Build a logistic model from data<\/li>\n<\/ul>\n<\/section>\n<h2>Build an Exponential Model from Data<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Exponential Regression:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used when data grows or decays at an increasing rate<\/li>\n<li class=\"whitespace-normal break-words\">Model form: [latex]y = ab^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]b > 1[\/latex]: exponential growth<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]0 < b < 1[\/latex]: exponential decay<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Using a Graphing Calculator:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Enter data in lists L1 (x-values) and L2 (y-values)<\/li>\n<li class=\"whitespace-normal break-words\">Create a scatter plot to verify exponential pattern<\/li>\n<li class=\"whitespace-normal break-words\">Use &#8220;ExpReg&#8221; from STAT CALC menu<\/li>\n<li class=\"whitespace-normal break-words\">Record values for [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Using Technology (Desmos):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Create a table with your data<\/li>\n<li class=\"whitespace-normal break-words\">Enter [latex]y_1 \\sim ab^{x_1}[\/latex] below the table<\/li>\n<li class=\"whitespace-normal break-words\">The tool displays values for [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">A credit card balance grows each month after graduation. The table shows the debt over 8 months:[latex]\\begin{array}{|c|c|c|c|c|c|c|c|c|}  \\hline  \\text{Month} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\  \\hline  \\text{Debt (\\$)} & 620 & 761.88 & 899.80 & 1039.93 & 1270.63 & 1589.04 & 1851.31 & 2154.92 \\\\  \\hline  \\end{array}[\/latex]Use exponential regression to fit a model to these data. If spending continues at this rate, what will the debt be after one year (month 12)?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qexp-reg-001\">Show Solution<\/button><\/p>\n<div id=\"qexp-reg-001\" class=\"hidden-answer\" style=\"display: none\">\nEnter the data into a graphing utility and use exponential regression.The model is: [latex]y = 522.89(1.196)^x[\/latex]To find the debt after 12 months:<br \/>\n[latex]\\begin{align*}  y &= 522.89(1.196)^x \\\\  y &= 522.89(1.196)^{12} \\\\  y &\\approx 4499.38  \\end{align*}[\/latex]The credit card debt will be approximately $4,499.38 after one year.\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox youChoose\" aria-label=\"You Choose\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Choose a Calculator<\/h3>\n<form><label>Select Calculator Type:<\/label><select name=\"video\"><option value=\"\/\/plugin.3playmedia.com\/show?mf=14660359&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=WpBmF3Ma0Hs&amp;video_target=tpm-plugin-drvnscs7-WpBmF3Ma0Hs\">TI-84 Plus CE<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf=14660360&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=O-hx7i3TkEg&amp;video_target=tpm-plugin-drvnscs7-O-hx7i3TkEg\">Desmos<\/option><\/select><\/form>\n<div class=\"videoContainer threePlay\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=14660359&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=WpBmF3Ma0Hs&amp;video_target=tpm-plugin-drvnscs7-WpBmF3Ma0Hs\" allowfullscreen><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+do+Exponential+Regression+on+the+TI84+Plus+CE_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to do Exponential Regression on the TI84 Plus CE\u201d here (opens in new window).<\/a><br \/>\nYou can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Exponential+Regression+-+Desmos_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential Regression &#8211; Desmos\u201d here (opens in new window).<\/a><\/section>\n<h2>Build a Logarithmic Model from Data<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Logarithmic Regression:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used when data increases or decreases rapidly at first, then slows<\/li>\n<li class=\"whitespace-normal break-words\">Model form: [latex]y = a + b\\ln(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">All input values must be positive<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]b > 0[\/latex]: increasing model<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]b < 0[\/latex]: decreasing model<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Using a Graphing Calculator:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Enter data in lists L1 and L2<\/li>\n<li class=\"whitespace-normal break-words\">Verify scatter plot shows logarithmic pattern<\/li>\n<li class=\"whitespace-normal break-words\">Use &#8220;LnReg&#8221; from STAT CALC menu<\/li>\n<li class=\"whitespace-normal break-words\">Record model as [latex]y = a + b\\ln(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Life expectancy has been increasing since 1900. The table shows average life expectancies of Americans:[latex]\\begin{array}{|c|c|c|c|c|c|c|}  \\hline  \\text{Year} & 1900 & 1910 & 1920 & 1930 & 1940 & 1950 \\\\  \\hline  \\text{Life Exp.} & 47.3 & 50.0 & 54.1 & 59.7 & 62.9 & 68.2 \\\\  \\hline  \\end{array}[\/latex][latex]\\begin{array}{|c|c|c|c|c|c|c|}  \\hline  \\text{Year} & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\\\  \\hline  \\text{Life Exp.} & 69.7 & 70.8 & 73.7 & 75.4 & 76.8 & 78.7 \\\\  \\hline  \\end{array}[\/latex]Let [latex]x = 1[\/latex] for 1900, [latex]x = 2[\/latex] for 1910, etc. Use logarithmic regression to predict life expectancy in 2030.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qlog-reg-001\">Show Solution<\/button><\/p>\n<div id=\"qlog-reg-001\" class=\"hidden-answer\" style=\"display: none\">\nEnter the data using [latex]x[\/latex] values 1\u201312 and use logarithmic regression.The model is: [latex]y = 42.53 + 13.86\\ln(x)[\/latex]For 2030, use [latex]x = 14[\/latex]:<br \/>\n[latex]\\begin{align*}  y &= 42.53 + 13.86\\ln(14) \\\\  y &\\approx 79.1  \\end{align*}[\/latex]The predicted life expectancy for 2030 is approximately 79.1 years.\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Choose a Calculator<\/h3>\n<form><label>Select Calculator:<\/label><select name=\"video\"><option value=\"\/\/plugin.3playmedia.com\/show?mf=14660361&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=YCRgsUSotEY&amp;video_target=tpm-plugin-drvnscs7-YCRgsUSotEY\">TI-84+<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf=14660362&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pqM1hzHMx9g&amp;video_target=tpm-plugin-drvnscs7-pqM1hzHMx9g\">Desmos<\/option><\/select><\/form>\n<div class=\"videoContainer threePlay\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=14660361&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=YCRgsUSotEY&amp;video_target=tpm-plugin-drvnscs7-YCRgsUSotEY\" allowfullscreen><\/iframe><\/div>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Logarithmic+Regression+on+the+TI84_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogarithmic Regression on the TI84\u201d here (opens in new window).<\/a><br \/>\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Desmos+Logarithmic+Regression_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDesmos Logarithmic Regression\u201d here (opens in new window).<\/a><\/section>\n<h2>Build a Logistic Model from Data<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Logistic Regression:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used when growth starts rapidly then levels off to an upper limit<\/li>\n<li class=\"whitespace-normal break-words\">Model form: [latex]y = \\frac{c}{1 + ae^{-bx}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]c[\/latex] is the limiting value (carrying capacity)<\/li>\n<li class=\"whitespace-normal break-words\">Initial value is [latex]\\frac{c}{1+a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Using a Graphing Calculator:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Enter data in lists L1 and L2<\/li>\n<li class=\"whitespace-normal break-words\">Verify scatter plot shows logistic pattern (S-shaped curve)<\/li>\n<li class=\"whitespace-normal break-words\">Use &#8220;Logistic&#8221; from STAT CALC menu<\/li>\n<li class=\"whitespace-normal break-words\">Record values for [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Harbor seal populations in the Wadden Sea from 1997-2012:[latex]\\begin{array}{|c|c|c|c|c|}  \\hline  \\text{Year} & 1997 & 1998 & 1999 & 2000 \\\\  \\hline  \\text{Population (thousands)} & 3.493 & 5.282 & 6.357 & 9.201 \\\\  \\hline  \\end{array}[\/latex](data continues through 2012 with population reaching 25.108)Let [latex]x = 0[\/latex] for 1997. Use logistic regression to:<br \/>\na. Fit a model to the data<br \/>\nb. Predict seal population in 2020<br \/>\nc. Find the carrying capacity<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qlog-reg-002\">Show Solution<\/button><\/p>\n<div id=\"qlog-reg-002\" class=\"hidden-answer\" style=\"display: none\">\nUsing logistic regression:a. The model is: [latex]y = \\frac{25.66}{1 + 6.11e^{-0.385x}}[\/latex]b. For 2020, use [latex]x = 23[\/latex]:<br \/>\n[latex]y = \\frac{25.66}{1 + 6.11e^{-0.385(23)}} \\approx 25.63[\/latex]About 25,634 seals in 2020.c. The carrying capacity is [latex]c \\approx 25,657[\/latex] seals.\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Choose a Calculator<\/h3>\n<form><label>Select Calculator:<\/label><select name=\"video\"><option value=\"\/\/plugin.3playmedia.com\/show?mf=14660363&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fRiTk2ywiio&amp;video_target=tpm-plugin-drvnscs7-fRiTk2ywiio\">TI-84+<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf=14679710&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=b3LsIcZzrHI&amp;video_target=tpm-plugin-drvnscs7-b3LsIcZzrHI\">Desmos<\/option><\/select><\/form>\n<div class=\"videoContainer threePlay\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=14660363&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fRiTk2ywiio&amp;video_target=tpm-plugin-drvnscs7-fRiTk2ywiio\" allowfullscreen><\/iframe><\/div>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Logistic+Regression+on+the+TI84_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogistic Regression on the TI84\u201d here (opens in new window).<\/a><br \/>\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/5+7+Logistical+Regression_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c5 7 Logistical Regression\u201d here (opens in new window).<\/a><\/section>\n","protected":false},"author":67,"menu_order":29,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"How to do Exponential Regression on the TI84 Plus CE\",\"author\":\"Robert Wilson\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/WpBmF3Ma0Hs\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Exponential Regression - Desmos\",\"author\":\"Susan Elliott\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/O-hx7i3TkEg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Logarithmic Regression on the TI84\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/YCRgsUSotEY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Desmos Logarithmic Regression\",\"author\":\"Debbie Gaydos\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/pqM1hzHMx9g\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Logistic Regression on the TI84\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/fRiTk2ywiio\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"5 7 Logistical Regression\",\"author\":\"\",\"organization\":\"MJC Math - YP\",\"url\":\"https:\/\/youtu.be\/b3LsIcZzrHI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":510,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"How to do Exponential Regression on the TI84 Plus CE","author":"Robert Wilson","organization":"","url":"https:\/\/youtu.be\/WpBmF3Ma0Hs","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Exponential Regression - Desmos","author":"Susan Elliott","organization":"","url":"https:\/\/youtu.be\/O-hx7i3TkEg","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Logarithmic Regression on the TI84","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/YCRgsUSotEY","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Desmos Logarithmic Regression","author":"Debbie Gaydos","organization":"","url":"https:\/\/youtu.be\/pqM1hzHMx9g","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Logistic Regression on the TI84","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/fRiTk2ywiio","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"5 7 Logistical Regression","author":"","organization":"MJC Math - YP","url":"https:\/\/youtu.be\/b3LsIcZzrHI","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":[{"divId":"tnh-video-picker","title":"Choose a Calculator","label":"Select Calculator Type","video_collection":[{"displayName":"TI-84 Plus CE","value":"\/\/plugin.3playmedia.com\/show?mf=14660359&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=WpBmF3Ma0Hs&amp;video_target=tpm-plugin-drvnscs7-WpBmF3Ma0Hs"},{"displayName":"Desmos","value":"\/\/plugin.3playmedia.com\/show?mf=14660360&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=O-hx7i3TkEg&amp;video_target=tpm-plugin-drvnscs7-O-hx7i3TkEg"}]},{"divId":"tnh-video-picker","title":"Choose a Calculator","label":"Select Calculator","video_collection":[{"displayName":"TI-84+","value":"\/\/plugin.3playmedia.com\/show?mf=14660361&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=YCRgsUSotEY&amp;video_target=tpm-plugin-drvnscs7-YCRgsUSotEY"},{"displayName":"Desmos","value":"\/\/plugin.3playmedia.com\/show?mf=14660362&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pqM1hzHMx9g&amp;video_target=tpm-plugin-drvnscs7-pqM1hzHMx9g"}]},{"divId":"tnh-video-picker","title":"Choose a Calculator","label":"Select Calculator","video_collection":[{"displayName":"TI-84+","value":"\/\/plugin.3playmedia.com\/show?mf=14660363&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=fRiTk2ywiio&amp;video_target=tpm-plugin-drvnscs7-fRiTk2ywiio"},{"displayName":"Desmos","value":"\/\/plugin.3playmedia.com\/show?mf=14679710&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=b3LsIcZzrHI&amp;video_target=tpm-plugin-drvnscs7-b3LsIcZzrHI"}]}],"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1449"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":23,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1449\/revisions"}],"predecessor-version":[{"id":5784,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1449\/revisions\/5784"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/510"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1449\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1449"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1449"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1449"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1449"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}