{"id":2083,"date":"2016-11-03T17:07:37","date_gmt":"2016-11-03T17:07:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2083"},"modified":"2017-04-04T19:35:16","modified_gmt":"2017-04-04T19:35:16","slug":"why-it-matters-exponential-and-logarithmic-equations-and-models","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/why-it-matters-exponential-and-logarithmic-equations-and-models\/","title":{"raw":"Why It Matters: Exponential and Logarithmic Equations and Models","rendered":"Why It Matters: Exponential and Logarithmic Equations and Models"},"content":{"raw":"<h2>Where are applications of exponential and logarithmic functions found?<\/h2>\r\n<img class=\"aligncenter wp-image-3738\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/03\/17190410\/fossil-1000575_1920-300x200.jpg\" alt=\"Image shows a man\u2019s hand holding a fossil outside near water.\" width=\"407\" height=\"271\" \/>\r\nYou\u2019ve gotten a job assisting an archeologist who has just discovered a fossil that appears to be an animal bone. \u00a0She assigns you the task of determining how old the bone is. \u00a0Where do you start?\r\n\r\nFortunately, you know that living things contain a radioactive form of carbon called carbon-14. \u00a0And like all radioactive elements, carbon-14 decays at a predictable rate known as the half-life. \u00a0The half-life of a radioactive element is the amount of time required for half of a sample to decay. \u00a0The half-life of carbon-14 is 5,730 years. \u00a0Given an original sample of carbon-14 is 100g, the table shows the mass remaining after each half-life.\r\n<div>\r\n<table style=\"width: 70%;\">\r\n<tbody>\r\n<tr>\r\n<td>Amount of sample (g)<\/td>\r\n<td>100<\/td>\r\n<td>50<\/td>\r\n<td>25<\/td>\r\n<td>12.5<\/td>\r\n<td>6.25<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Time (years)<\/td>\r\n<td>0<\/td>\r\n<td>5730<\/td>\r\n<td>11,460<\/td>\r\n<td>17,190<\/td>\r\n<td>22,920<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n\r\nBut, what if the bone started with a different mass of carbon-14 or a different number of years has passed? \u00a0To better study the bone, you need to know that the rate of decay of a radioactive element can be modeled with an exponential function. \u00a0Given a couple of data points, you can build a model that represents the decay of carbon over time for your specimen.\r\n\r\nAs you complete this module, keep the following questions in mind. \u00a0Then at the end of the module, we\u2019ll return to the develop a model for the decay of carbon-14.\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">How do you develop a model for the decay of carbon-14?<\/li>\r\n \t<li style=\"font-weight: 400;\">How can you use the model to determine the amount of carbon-14 that remains after any number of years?<\/li>\r\n \t<li style=\"font-weight: 400;\">What would a graph of the decay of carbon-14 look like?<\/li>\r\n<\/ul>\r\n&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<p class=\"p1\"><span class=\"s1\">Logarithmic Properties<\/span><\/p>\r\n\r\n<ul class=\"ul1\">\r\n \t<li>Use power, product, and quotient rules to expand and condense logarithms<\/li>\r\n \t<li>Use the change-of-base formula for logarithms<\/li>\r\n<\/ul>\r\n<p class=\"p1\"><span class=\"s1\">Exponential and Logarithmic\u00a0Equations<\/span><\/p>\r\n\r\n<ul class=\"ul1\">\r\n \t<li>Use like bases to solve exponential equations<\/li>\r\n \t<li>Use logarithms to solve exponential equations<\/li>\r\n \t<li>Use the definition of a logarithm to solve logarithmic equations<\/li>\r\n \t<li>Use the one-to-one property of logarithms to solve logarithmic equations<\/li>\r\n \t<li>Solve applied problems involving exponential and logarithmic equations<\/li>\r\n<\/ul>\r\n<p class=\"p1\"><span class=\"s1\">Exponential and Logarithmic Models<\/span><\/p>\r\n\r\n<ul class=\"ul1\">\r\n \t<li>Model exponential growth and decay<\/li>\r\n \t<li>Use Newton\u2019s Law of Cooling<\/li>\r\n \t<li>Use logistic-growth models<\/li>\r\n \t<li>Choose an appropriate model for data<\/li>\r\n \t<li>Express an exponential model in base <span class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: inherit; margin: 0px; padding: 1px 0px; border: 0px; outline: 0px; font-size: 17.44px; vertical-align: baseline; background: transparent; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;e&lt;\/mi&gt;&lt;\/math&gt;\"><span class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: inherit; margin: 0px; padding: 1px 0px; border: 0px; outline: 0px; font-size: 17.44px; vertical-align: baseline; background: transparent; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;e&lt;\/mi&gt;&lt;\/math&gt;\"><span class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: inherit; margin: 0px; padding: 1px 0px; border: 0px; outline: 0px; font-size: 17.44px; vertical-align: baseline; background: transparent; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;e&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MJXc-Node-19\" class=\"mjx-math\"><span id=\"MJXc-Node-20\" class=\"mjx-mrow\"><span id=\"MJXc-Node-21\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">e<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\">\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>e<\/mi><\/math><\/span><\/span><\/span><\/span><\/li>\r\n \t<li>Build an exponential model from data<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;","rendered":"<h2>Where are applications of exponential and logarithmic functions found?<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3738\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/03\/17190410\/fossil-1000575_1920-300x200.jpg\" alt=\"Image shows a man\u2019s hand holding a fossil outside near water.\" width=\"407\" height=\"271\" \/><br \/>\nYou\u2019ve gotten a job assisting an archeologist who has just discovered a fossil that appears to be an animal bone. \u00a0She assigns you the task of determining how old the bone is. \u00a0Where do you start?<\/p>\n<p>Fortunately, you know that living things contain a radioactive form of carbon called carbon-14. \u00a0And like all radioactive elements, carbon-14 decays at a predictable rate known as the half-life. \u00a0The half-life of a radioactive element is the amount of time required for half of a sample to decay. \u00a0The half-life of carbon-14 is 5,730 years. \u00a0Given an original sample of carbon-14 is 100g, the table shows the mass remaining after each half-life.<\/p>\n<div>\n<table style=\"width: 70%;\">\n<tbody>\n<tr>\n<td>Amount of sample (g)<\/td>\n<td>100<\/td>\n<td>50<\/td>\n<td>25<\/td>\n<td>12.5<\/td>\n<td>6.25<\/td>\n<\/tr>\n<tr>\n<td>Time (years)<\/td>\n<td>0<\/td>\n<td>5730<\/td>\n<td>11,460<\/td>\n<td>17,190<\/td>\n<td>22,920<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<p>But, what if the bone started with a different mass of carbon-14 or a different number of years has passed? \u00a0To better study the bone, you need to know that the rate of decay of a radioactive element can be modeled with an exponential function. \u00a0Given a couple of data points, you can build a model that represents the decay of carbon over time for your specimen.<\/p>\n<p>As you complete this module, keep the following questions in mind. \u00a0Then at the end of the module, we\u2019ll return to the develop a model for the decay of carbon-14.<\/p>\n<ul>\n<li style=\"font-weight: 400;\">How do you develop a model for the decay of carbon-14?<\/li>\n<li style=\"font-weight: 400;\">How can you use the model to determine the amount of carbon-14 that remains after any number of years?<\/li>\n<li style=\"font-weight: 400;\">What would a graph of the decay of carbon-14 look like?<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p class=\"p1\"><span class=\"s1\">Logarithmic Properties<\/span><\/p>\n<ul class=\"ul1\">\n<li>Use power, product, and quotient rules to expand and condense logarithms<\/li>\n<li>Use the change-of-base formula for logarithms<\/li>\n<\/ul>\n<p class=\"p1\"><span class=\"s1\">Exponential and Logarithmic\u00a0Equations<\/span><\/p>\n<ul class=\"ul1\">\n<li>Use like bases to solve exponential equations<\/li>\n<li>Use logarithms to solve exponential equations<\/li>\n<li>Use the definition of a logarithm to solve logarithmic equations<\/li>\n<li>Use the one-to-one property of logarithms to solve logarithmic equations<\/li>\n<li>Solve applied problems involving exponential and logarithmic equations<\/li>\n<\/ul>\n<p class=\"p1\"><span class=\"s1\">Exponential and Logarithmic Models<\/span><\/p>\n<ul class=\"ul1\">\n<li>Model exponential growth and decay<\/li>\n<li>Use Newton\u2019s Law of Cooling<\/li>\n<li>Use logistic-growth models<\/li>\n<li>Choose an appropriate model for data<\/li>\n<li>Express an exponential model in base <span class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: inherit; margin: 0px; padding: 1px 0px; border: 0px; outline: 0px; font-size: 17.44px; vertical-align: baseline; background: transparent; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;e&lt;\/mi&gt;&lt;\/math&gt;\"><span class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: inherit; margin: 0px; padding: 1px 0px; border: 0px; outline: 0px; font-size: 17.44px; vertical-align: baseline; background: transparent; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;e&lt;\/mi&gt;&lt;\/math&gt;\"><span class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: inherit; margin: 0px; padding: 1px 0px; border: 0px; outline: 0px; font-size: 17.44px; vertical-align: baseline; background: transparent; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;e&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MJXc-Node-19\" class=\"mjx-math\"><span id=\"MJXc-Node-20\" class=\"mjx-mrow\"><span id=\"MJXc-Node-21\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">e<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>e<\/mi><\/math><\/span><\/span><\/span><\/span><\/li>\n<li>Build an exponential model from data<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\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-2083\">\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>Why It Matters: Exponential and Logarithmic Equations and Models. <strong>Authored 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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>fossil. <strong>Authored by<\/strong>: diego_torres. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/fossil-england-coast-ocean-1000575\/\">https:\/\/pixabay.com\/en\/fossil-england-coast-ocean-1000575\/<\/a>. <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":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Why It Matters: Exponential and Logarithmic Equations and Models\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"fossil\",\"author\":\"diego_torres\",\"organization\":\"\",\"url\":\"https:\/\/pixabay.com\/en\/fossil-england-coast-ocean-1000575\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"946c676f-5626-455b-bc4e-6fbedeaf3dc5","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2083","chapter","type-chapter","status-publish","hentry"],"part":2074,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2083","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2083\/revisions"}],"predecessor-version":[{"id":3940,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2083\/revisions\/3940"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2074"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2083\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/media?parent=2083"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2083"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2083"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/license?post=2083"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}