{"id":3425,"date":"2024-06-24T16:37:28","date_gmt":"2024-06-24T16:37:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3425"},"modified":"2025-08-17T23:02:17","modified_gmt":"2025-08-17T23:02:17","slug":"derivatives-of-exponential-and-logarithmic-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-of-exponential-and-logarithmic-functions-apply-it\/","title":{"raw":"Derivatives of Exponential and Logarithmic Functions: Apply It","rendered":"Derivatives of Exponential and Logarithmic Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Determine the derivatives of exponential and logarithmic functions<\/li>\r\n\t<li>Apply logarithmic differentiation to find derivatives<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>One of the most commonly described phenomena using exponential and logarithmic functions is growth.\u00a0 Both exponential and logarithmic functions increase without bound, but the rate at which that happens for each function is different.<\/p>\r\n\r\n[caption id=\"attachment_3974\" align=\"aligncenter\" width=\"1116\"]<img class=\"wp-image-3974 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522.png\" alt=\"Two graphs displayed side by side. The left graph shows &quot;Logarithmic Growth,&quot; characterized by a curve that rises quickly initially and then gradually levels off. The right graph shows &quot;Exponential Growth,&quot; with a curve that starts slowly and then rises rapidly, becoming steeper over time.\" width=\"1116\" height=\"423\" \/> Comparison of logarithmic growth and exponential growth[\/caption]\r\n\r\n<p>Exponential growth starts off slow and gets faster and faster.\u00a0 For instance, the amount of traffic a website gets is slow at first, but as more people visit, word spreads and search engines start recommending the site, the number of visits become greater and greater.\u00a0<\/p>\r\n<p>\u200b\u200bOn the other hand, logarithmic growth increases very quickly at the beginning but slows down as time goes on. For instance, when you begin a strength training journey you often see large gains right away, but as the journey goes on the gains become smaller or farther apart, so the rate slows down.\u00a0<\/p>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]287942[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]287943[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm_question hide_question_numbers=1]287946[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Determine the derivatives of exponential and logarithmic functions<\/li>\n<li>Apply logarithmic differentiation to find derivatives<\/li>\n<\/ul>\n<\/section>\n<p>One of the most commonly described phenomena using exponential and logarithmic functions is growth.\u00a0 Both exponential and logarithmic functions increase without bound, but the rate at which that happens for each function is different.<\/p>\n<figure id=\"attachment_3974\" aria-describedby=\"caption-attachment-3974\" style=\"width: 1116px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3974 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522.png\" alt=\"Two graphs displayed side by side. The left graph shows &quot;Logarithmic Growth,&quot; characterized by a curve that rises quickly initially and then gradually levels off. The right graph shows &quot;Exponential Growth,&quot; with a curve that starts slowly and then rises rapidly, becoming steeper over time.\" width=\"1116\" height=\"423\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522.png 1116w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522-300x114.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522-1024x388.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522-768x291.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522-65x25.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522-225x85.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16173536\/Screenshot-2024-07-16-133522-350x133.png 350w\" sizes=\"(max-width: 1116px) 100vw, 1116px\" \/><figcaption id=\"caption-attachment-3974\" class=\"wp-caption-text\">Comparison of logarithmic growth and exponential growth<\/figcaption><\/figure>\n<p>Exponential growth starts off slow and gets faster and faster.\u00a0 For instance, the amount of traffic a website gets is slow at first, but as more people visit, word spreads and search engines start recommending the site, the number of visits become greater and greater.\u00a0<\/p>\n<p>\u200b\u200bOn the other hand, logarithmic growth increases very quickly at the beginning but slows down as time goes on. For instance, when you begin a strength training journey you often see large gains right away, but as the journey goes on the gains become smaller or farther apart, so the rate slows down.\u00a0<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm287942\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287942&theme=lumen&iframe_resize_id=ohm287942&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm287943\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287943&theme=lumen&iframe_resize_id=ohm287943&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm287946\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287946&theme=lumen&iframe_resize_id=ohm287946&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":26,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":560,"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3425"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3425\/revisions"}],"predecessor-version":[{"id":4766,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3425\/revisions\/4766"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/560"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3425\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3425"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3425"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3425"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3425"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}