{"id":1238,"date":"2017-02-01T00:08:50","date_gmt":"2017-02-01T00:08:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1238"},"modified":"2017-04-18T19:35:16","modified_gmt":"2017-04-18T19:35:16","slug":"introduction-exponential-and-logistic-growth","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/chapter\/introduction-exponential-and-logistic-growth\/","title":{"raw":"Introduction: Logarithms and Logistic Growth","rendered":"Introduction: Logarithms and Logistic Growth"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Evaluate and rewrite logarithms using the properties of logarithms<\/li>\r\n \t<li>Use the properties of logarithms to solve exponential models\u00a0for time<\/li>\r\n \t<li>Identify the carrying capacity in a logistic growth model<\/li>\r\n \t<li>Use a logistic growth model to predict growth<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn a confined environment the growth rate of a population may not remain constant. In a lake, for example, there is some <em>maximum sustainable population<\/em> of fish, also called a <strong>carrying capacity<\/strong>. In this section, we will develop a model that contains a carrying capacity term, and use it to predict growth under constraints. \u00a0Because resources are typically limited in systems, these types of models are much more common than linear or geometric growth.\r\n\r\n[caption id=\"attachment_1279\" align=\"aligncenter\" width=\"534\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/02000229\/Mandel_zoom_08_satellite_antenna.jpg\"><img class=\"wp-image-1279\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/02000229\/Mandel_zoom_08_satellite_antenna-300x225.jpg\" alt=\"The famous Mandelbrot set, a fractal whose growth is constrained.\" width=\"534\" height=\"402\" \/><\/a> The famous Mandelbrot set, a fractal whose growth is constrained.[\/caption]","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Evaluate and rewrite logarithms using the properties of logarithms<\/li>\n<li>Use the properties of logarithms to solve exponential models\u00a0for time<\/li>\n<li>Identify the carrying capacity in a logistic growth model<\/li>\n<li>Use a logistic growth model to predict growth<\/li>\n<\/ul>\n<\/div>\n<p>In a confined environment the growth rate of a population may not remain constant. In a lake, for example, there is some <em>maximum sustainable population<\/em> of fish, also called a <strong>carrying capacity<\/strong>. In this section, we will develop a model that contains a carrying capacity term, and use it to predict growth under constraints. \u00a0Because resources are typically limited in systems, these types of models are much more common than linear or geometric growth.<\/p>\n<div id=\"attachment_1279\" style=\"width: 544px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/02000229\/Mandel_zoom_08_satellite_antenna.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1279\" class=\"wp-image-1279\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/02000229\/Mandel_zoom_08_satellite_antenna-300x225.jpg\" alt=\"The famous Mandelbrot set, a fractal whose growth is constrained.\" width=\"534\" height=\"402\" \/><\/a><\/p>\n<p id=\"caption-attachment-1279\" class=\"wp-caption-text\">The famous Mandelbrot set, a fractal whose growth is constrained.<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1238\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Introduction and Learning Objectives. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Introduction and Learning Objectives\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"d9ca10d5-d7eb-47fb-836b-ec2938b29587","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1238","chapter","type-chapter","status-publish","hentry"],"part":356,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters\/1238","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters\/1238\/revisions"}],"predecessor-version":[{"id":1377,"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters\/1238\/revisions\/1377"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/pressbooks\/v2\/parts\/356"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapters\/1238\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/wp\/v2\/media?parent=1238"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/pressbooks\/v2\/chapter-type?post=1238"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/wp\/v2\/contributor?post=1238"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-mathforliberalarts\/wp-json\/wp\/v2\/license?post=1238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}