{"id":2640,"date":"2025-08-13T18:19:43","date_gmt":"2025-08-13T18:19:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2640"},"modified":"2026-03-11T09:06:20","modified_gmt":"2026-03-11T09:06:20","slug":"exponential-and-logarithmic-equations-background-youll-need-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-equations-background-youll-need-4\/","title":{"raw":"Exponential and Logarithmic Equations: Background You'll Need 4","rendered":"Exponential and Logarithmic Equations: Background You&#8217;ll Need 4"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Use linear regression to model data<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-start=\"203\" data-end=\"242\">Use Linear Regression to Model Data<\/h2>\r\n<p data-start=\"244\" data-end=\"516\">When data is collected in the real world, it often does not line up perfectly along a straight line. However, the data may still follow a roughly linear pattern. In this case, we use a <strong data-start=\"429\" data-end=\"449\">line of best fit<\/strong> to model the data. This process is called <strong data-start=\"492\" data-end=\"513\">linear regression<\/strong>.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>linear regression<\/h3>\r\nLinear regression produces a line of best fit for data that is approximately linear. The equation has the form [latex]y = mx + b[\/latex] or [latex]y = a + bx[\/latex].\r\n\r\n&nbsp;\r\n\r\nThe regression line can be used for prediction\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">The line may not pass through every point (or even any point), but it minimizes the overall error (the distance from the points to the line).<\/section><section class=\"textbox interact\" aria-label=\"Interact\">[videopicker divId=\"tnh-video-picker\" title=\"Linear Regression with Technology\" label=\"Select Tool\"]\r\n[videooption displayName=\"Linear Regression with Desmos\" value=\"\/\/plugin.3playmedia.com\/show?mf=14660357&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VfC8uQGm5W8&amp;video_target=tpm-plugin-drvnscs7-VfC8uQGm5W8\"][videooption displayName=\"Linear Regression with Calculator\" value=\"\/\/plugin.3playmedia.com\/show?mf= 14660358&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=0as2Jh_eDwg&amp;video_target=tpm-plugin-drvnscs7-0as2Jh_eDwg\"][\/videopicker]You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Learn+Desmos+-+Regressions+%5B2015%5D_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLearn Desmos: Regressions [2015]\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\/Linear+Regression+TI84+(Line+of+Best+Fit)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear Regression TI84 (Line of Best Fit)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"260\" data-end=\"378\">A company tracks the number of social media ads run each month and the corresponding sales. The data is shown below:<\/p>\r\n\r\n<div class=\"_tableContainer_1rjym_1\">\r\n<div class=\"_tableWrapper_1rjym_13 group flex w-fit flex-col-reverse\" tabindex=\"-1\">\r\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"380\" data-end=\"573\">\r\n<thead data-start=\"380\" data-end=\"403\">\r\n<tr data-start=\"380\" data-end=\"403\">\r\n<th data-start=\"380\" data-end=\"390\" data-col-size=\"sm\">Ads (x)<\/th>\r\n<th data-start=\"390\" data-end=\"403\" data-col-size=\"sm\">Sales (y)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody data-start=\"428\" data-end=\"573\">\r\n<tr data-start=\"428\" data-end=\"451\">\r\n<td data-start=\"428\" data-end=\"438\" data-col-size=\"sm\">2<\/td>\r\n<td data-col-size=\"sm\" data-start=\"438\" data-end=\"451\">7<\/td>\r\n<\/tr>\r\n<tr data-start=\"452\" data-end=\"475\">\r\n<td data-start=\"452\" data-end=\"462\" data-col-size=\"sm\">4<\/td>\r\n<td data-col-size=\"sm\" data-start=\"462\" data-end=\"475\">9<\/td>\r\n<\/tr>\r\n<tr data-start=\"476\" data-end=\"499\">\r\n<td data-start=\"476\" data-end=\"486\" data-col-size=\"sm\">6<\/td>\r\n<td data-col-size=\"sm\" data-start=\"486\" data-end=\"499\">14<\/td>\r\n<\/tr>\r\n<tr data-start=\"500\" data-end=\"523\">\r\n<td data-start=\"500\" data-end=\"510\" data-col-size=\"sm\">8<\/td>\r\n<td data-col-size=\"sm\" data-start=\"510\" data-end=\"523\">17<\/td>\r\n<\/tr>\r\n<tr data-start=\"524\" data-end=\"547\">\r\n<td data-start=\"524\" data-end=\"534\" data-col-size=\"sm\">10<\/td>\r\n<td data-col-size=\"sm\" data-start=\"534\" data-end=\"547\">21<\/td>\r\n<\/tr>\r\n<tr data-start=\"548\" data-end=\"573\">\r\n<td data-start=\"548\" data-end=\"558\" data-col-size=\"sm\">12<\/td>\r\n<td data-col-size=\"sm\" data-start=\"558\" data-end=\"573\">24<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<p data-start=\"575\" data-end=\"682\">Find a linear regression model for the data. Then use the model to predict sales if the company runs 15 ads.<\/p>\r\n<p data-start=\"575\" data-end=\"682\">[reveal-answer q=\"78400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"78400\"]The points do not line up perfectly, but they follow a roughly linear trend. Using technology, the regression line is found to be [latex]y = 1.7x + 2.9[\/latex] To make a prediction for 15 ads: [latex]\\begin{aligned} y &amp;= 1.7(15) + 2.9 \\[6pt] y &amp;= 25.5 + 2.9 \\[6pt] y &amp;= 28.4 \\end{aligned}[\/latex] So the model predicts about 28 sales.[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311975[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use linear regression to model data<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 data-start=\"203\" data-end=\"242\">Use Linear Regression to Model Data<\/h2>\n<p data-start=\"244\" data-end=\"516\">When data is collected in the real world, it often does not line up perfectly along a straight line. However, the data may still follow a roughly linear pattern. In this case, we use a <strong data-start=\"429\" data-end=\"449\">line of best fit<\/strong> to model the data. This process is called <strong data-start=\"492\" data-end=\"513\">linear regression<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>linear regression<\/h3>\n<p>Linear regression produces a line of best fit for data that is approximately linear. The equation has the form [latex]y = mx + b[\/latex] or [latex]y = a + bx[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>The regression line can be used for prediction<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The line may not pass through every point (or even any point), but it minimizes the overall error (the distance from the points to the line).<\/section>\n<section class=\"textbox interact\" aria-label=\"Interact\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Linear Regression with Technology<\/h3>\n<form><label>Select Tool:<\/label><select name=\"video\"><option value=\"\/\/plugin.3playmedia.com\/show?mf=14660357&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VfC8uQGm5W8&amp;video_target=tpm-plugin-drvnscs7-VfC8uQGm5W8&#8243;\">Linear Regression with Desmos<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf= 14660358&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=0as2Jh_eDwg&amp;video_target=tpm-plugin-drvnscs7-0as2Jh_eDwg\">Linear Regression with Calculator<\/option><\/select><\/form>\n<div class=\"videoContainer threePlay\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=14660357&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VfC8uQGm5W8&amp;video_target=tpm-plugin-drvnscs7-VfC8uQGm5W8&#8243;\" allowfullscreen><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Learn+Desmos+-+Regressions+%5B2015%5D_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLearn Desmos: Regressions [2015]\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\/Linear+Regression+TI84+(Line+of+Best+Fit)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLinear Regression TI84 (Line of Best Fit)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"260\" data-end=\"378\">A company tracks the number of social media ads run each month and the corresponding sales. The data is shown below:<\/p>\n<div class=\"_tableContainer_1rjym_1\">\n<div class=\"_tableWrapper_1rjym_13 group flex w-fit flex-col-reverse\" tabindex=\"-1\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"380\" data-end=\"573\">\n<thead data-start=\"380\" data-end=\"403\">\n<tr data-start=\"380\" data-end=\"403\">\n<th data-start=\"380\" data-end=\"390\" data-col-size=\"sm\">Ads (x)<\/th>\n<th data-start=\"390\" data-end=\"403\" data-col-size=\"sm\">Sales (y)<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"428\" data-end=\"573\">\n<tr data-start=\"428\" data-end=\"451\">\n<td data-start=\"428\" data-end=\"438\" data-col-size=\"sm\">2<\/td>\n<td data-col-size=\"sm\" data-start=\"438\" data-end=\"451\">7<\/td>\n<\/tr>\n<tr data-start=\"452\" data-end=\"475\">\n<td data-start=\"452\" data-end=\"462\" data-col-size=\"sm\">4<\/td>\n<td data-col-size=\"sm\" data-start=\"462\" data-end=\"475\">9<\/td>\n<\/tr>\n<tr data-start=\"476\" data-end=\"499\">\n<td data-start=\"476\" data-end=\"486\" data-col-size=\"sm\">6<\/td>\n<td data-col-size=\"sm\" data-start=\"486\" data-end=\"499\">14<\/td>\n<\/tr>\n<tr data-start=\"500\" data-end=\"523\">\n<td data-start=\"500\" data-end=\"510\" data-col-size=\"sm\">8<\/td>\n<td data-col-size=\"sm\" data-start=\"510\" data-end=\"523\">17<\/td>\n<\/tr>\n<tr data-start=\"524\" data-end=\"547\">\n<td data-start=\"524\" data-end=\"534\" data-col-size=\"sm\">10<\/td>\n<td data-col-size=\"sm\" data-start=\"534\" data-end=\"547\">21<\/td>\n<\/tr>\n<tr data-start=\"548\" data-end=\"573\">\n<td data-start=\"548\" data-end=\"558\" data-col-size=\"sm\">12<\/td>\n<td data-col-size=\"sm\" data-start=\"558\" data-end=\"573\">24<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p data-start=\"575\" data-end=\"682\">Find a linear regression model for the data. Then use the model to predict sales if the company runs 15 ads.<\/p>\n<p data-start=\"575\" data-end=\"682\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q78400\">Show Solution<\/button><\/p>\n<div id=\"q78400\" class=\"hidden-answer\" style=\"display: none\">The points do not line up perfectly, but they follow a roughly linear trend. Using technology, the regression line is found to be [latex]y = 1.7x + 2.9[\/latex] To make a prediction for 15 ads: [latex]\\begin{aligned} y &= 1.7(15) + 2.9 \\[6pt] y &= 25.5 + 2.9 \\[6pt] y &= 28.4 \\end{aligned}[\/latex] So the model predicts about 28 sales.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311975\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311975&theme=lumen&iframe_resize_id=ohm311975&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Learn Desmos: Regressions [2015]\",\"author\":\"\",\"organization\":\"Desmos\",\"url\":\"https:\/\/youtu.be\/VfC8uQGm5W8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Linear Regression TI84 (Line of Best Fit)\",\"author\":\"\",\"organization\":\"Mario\\'s Math Tutoring\",\"url\":\"https:\/\/youtu.be\/0as2Jh_eDwg\",\"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":"background_you_need","content_attributions":null,"internal_book_links":[],"video_content":[{"divId":"tnh-video-picker","title":"Linear Regression with Technology","label":"Select Tool","video_collection":[{"displayName":"Linear Regression with Desmos","value":"\/\/plugin.3playmedia.com\/show?mf=14660357&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VfC8uQGm5W8&amp;video_target=tpm-plugin-drvnscs7-VfC8uQGm5W8"},{"displayName":"Linear Regression with Calculator","value":"\/\/plugin.3playmedia.com\/show?mf= 14660358&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=0as2Jh_eDwg&amp;video_target=tpm-plugin-drvnscs7-0as2Jh_eDwg"}]}],"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\/2640"}],"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":11,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2640\/revisions"}],"predecessor-version":[{"id":5851,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2640\/revisions\/5851"}],"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\/2640\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2640"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2640"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2640"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2640"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}