{"id":3173,"date":"2025-08-15T22:27:38","date_gmt":"2025-08-15T22:27:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3173"},"modified":"2025-09-16T16:01:29","modified_gmt":"2025-09-16T16:01:29","slug":"fitting-exponential-models-to-data-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/fitting-exponential-models-to-data-apply-it\/","title":{"raw":"Fitting Exponential Models to Data: Apply It","rendered":"Fitting Exponential Models to Data: Apply It"},"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 class=\"text-2xl font-bold mt-1 text-text-100\">Building Exponential Models From Data<\/h2>\r\n<p class=\"whitespace-normal break-words\">A tech startup is tracking the growth of their mobile app downloads over the first 8 months after launch. The marketing team needs to predict future download patterns to plan their server capacity and advertising budget.<\/p>\r\n\r\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\">\r\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">Month<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">Downloads (thousands)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">1<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">12.5<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">2<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">18.7<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">3<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">28.1<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">4<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">42.3<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">5<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">63.8<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">6<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">96.2<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">7<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">145.1<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">8<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">218.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312083[\/ohm_question]<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">In exponential growth, you're looking for data that increases by multiplication rather than addition.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312087[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Type your <em>Try It<\/em> text here<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\">Mathematical models are tools that work well within reasonable ranges, but real-world factors often create limits that pure exponential growth doesn't account for. This is why models like logistic growth (which includes a limiting factor) are sometimes more realistic for long-term predictions.<\/p>\r\n\r\n<\/section>\r\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Building Logarithmic Models From Data<\/h2>\r\n<p class=\"whitespace-normal break-words\">As the startup gained more downloads, they focused on improving user experience based on feedback. They implemented the most requested features first, leading to big jumps in satisfaction. However, as they addressed major issues, remaining improvements became smaller and harder to achieve.<\/p>\r\n\r\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\">\r\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">Month<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">Satisfaction Score (1-100)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">1<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">45<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">2<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">62<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">3<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">71<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">4<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">76<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">5<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">79<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">6<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">82<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312097 [\/ohm_question]<\/section><section class=\"textbox recall\" aria-label=\"Recall\">Logarithmic growth starts fast but slows down over time.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312098[\/ohm_question]<\/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 class=\"text-2xl font-bold mt-1 text-text-100\">Building Exponential Models From Data<\/h2>\n<p class=\"whitespace-normal break-words\">A tech startup is tracking the growth of their mobile app downloads over the first 8 months after launch. The marketing team needs to predict future download patterns to plan their server capacity and advertising budget.<\/p>\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\">\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">Month<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">Downloads (thousands)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">1<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">12.5<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">2<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">18.7<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">3<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">28.1<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">4<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">42.3<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">5<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">63.8<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">6<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">96.2<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">7<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">145.1<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">8<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">218.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312083\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312083&theme=lumen&iframe_resize_id=ohm312083&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">In exponential growth, you&#8217;re looking for data that increases by multiplication rather than addition.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312087\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312087&theme=lumen&iframe_resize_id=ohm312087&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Type your <em>Try It<\/em> text here<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\">Mathematical models are tools that work well within reasonable ranges, but real-world factors often create limits that pure exponential growth doesn&#8217;t account for. This is why models like logistic growth (which includes a limiting factor) are sometimes more realistic for long-term predictions.<\/p>\n<\/section>\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Building Logarithmic Models From Data<\/h2>\n<p class=\"whitespace-normal break-words\">As the startup gained more downloads, they focused on improving user experience based on feedback. They implemented the most requested features first, leading to big jumps in satisfaction. However, as they addressed major issues, remaining improvements became smaller and harder to achieve.<\/p>\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\">\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">Month<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">Satisfaction Score (1-100)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">1<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">45<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">2<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">62<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">3<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">71<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">4<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">76<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">5<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">79<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">6<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\">82<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312097\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312097&theme=lumen&iframe_resize_id=ohm312097&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">Logarithmic growth starts fast but slows down over time.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312098\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312098&theme=lumen&iframe_resize_id=ohm312098&source=tnh\" width=\"100%\" 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