{"id":93,"date":"2017-08-23T07:48:01","date_gmt":"2017-08-23T07:48:01","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=93"},"modified":"2017-08-30T06:01:42","modified_gmt":"2017-08-30T06:01:42","slug":"exponential-functions","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/exponential-functions\/","title":{"rendered":"Exponential Functions"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/laws-of-fractional-exponents\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Next Workshop \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Exponential Functions<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study the standard form of an exponential function and behavior of growth and decay in\u00a0functions. You will also review how to solve equations in exponential form (the variable in the equation is located\u00a0in the exponent).<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li><strong><em>Fractional\u00a0exponents<\/em><\/strong> obey the same rules as integer exponents,\u00a0with regards to multiplying, dividing, adding, and subtracting.<\/li>\n<li><em><strong>Radicals<\/strong><\/em> are a form of\u00a0fractional exponent.<\/li>\n<\/ul>\n<section>\n<h3>What is an exponential function?<\/h3>\n<p>The basic form of an exponential function is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p2_clip_image003.gif\" width=\"131\" height=\"15\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>The variables are defined as:<\/p>\n<ul>\n<li><em>a\u00a0<\/em>is a constant,<\/li>\n<li><em>b\u00a0<\/em>is the base, and<\/li>\n<li><em>x\u00a0<\/em>is the exponent.<\/li>\n<\/ul>\n<h4>Exponential Growth<\/h4>\n<p>When <em>b\u00a0<\/em>&gt; 1, the function has exponential growth. An example of a\u00a0growth function model is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p2_clip_image006.gif\" width=\"56\" height=\"15\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This example is graphed below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math-Mod-3.3-Art-001.gif\" alt=\" Exponential growth\" width=\"309\" height=\"261\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h4>Exponential Decay<\/h4>\n<p>When 0 &lt;\u00a0<em>b<\/em> &lt; 1, the\u00a0function has exponential decay. An example of a decay\u00a0function model is\u00a0<img decoding=\"async\" class=\"no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/s9_p2_html_m43160118.gif\" name=\"graphics6\" \/>.\u00a0This example is graphed below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/Math-Mod-3.3-Art-002.gif\" alt=\"Exponential decay\" width=\"316\" height=\"265\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which equation models exponential decay?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p2_clip_image012.gif\" width=\"56\" height=\"15\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p2_clip_image015.gif\" width=\"90\" height=\"15\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p2_clip_image018.gif\" width=\"49\" height=\"15\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p2_clip_image021.gif\" width=\"46\" height=\"15\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B.\u00a0For a function to decay, the base is between 0 &gt; <em>b\u00a0<\/em>&gt; 1. Choice B has a base of 0.75.<\/p>\n<\/div>\n<\/section>\n<p>The equation for compound\u00a0interest is an excellent example of exponential growth. It is\u00a0given by\u00a0<img decoding=\"async\" class=\"no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p2_clip_image024.gif\" name=\"graphics12\" \/>,\u00a0where <em>P =\u00a0<\/em>principle, <em>r\u00a0= <\/em>interest rate, <em>n\u00a0= c<\/em>ompounding\u00a0periods per year, <em>t\u00a0= <\/em>time, and <em>A\u00a0= <\/em>final amount.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>An initial investment of $5,000.00, compounded yearly at an\u00a0interest rate of 15%, will be worth what amount in 20 years?<\/p>\n<ol>\n<li>$5,806.23<\/li>\n<li>$6,500.40<\/li>\n<li>$58,506.20<\/li>\n<li>$81,832.69<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D.\u00a0The interest rate, in decimal form, is <em>r\u00a0<\/em>= 0.15.\u00a0Compounding is yearly, therefore <em>n\u00a0<\/em>= 1. The duration\u00a0is <em>t = <\/em>20\u00a0years.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p2_clip_image027.gif\" width=\"253\" height=\"49\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3>How are logarithms used in exponential functions?<\/h3>\n<p>To solve exponential functions, we must first introduce the\u00a0concept of <abbr title=\" the exponent that indicates the power to which a number is raised to produce another given number\">logarithms<\/abbr>.\u00a0The definition of a logarithm is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p3_clip_image003.gif\" width=\"205\" height=\"17\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>Logarithms are very useful\u00a0in solving exponential equations. Using logarithms, we now have a\u00a0way to transfer the variable <em>x\u00a0<\/em>out of the exponential position. This allows us to solve the\u00a0equation.<\/p>\n<p>Let&#8217;s try an example. Write the exponential equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p3_clip_image006.gif\" width=\"44\" height=\"12\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>\u00a0in logarithmic form.<\/p>\n<p><strong>Step 1:<\/strong> Notice\u00a0the position of each number and the variable <em>x\u00a0<\/em>in the exponential equation.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p3_clip_image009.gif\" width=\"41\" height=\"38\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 2:<\/strong> Rewrite using the logarithmic\u00a0equation.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p3_clip_image012.gif\" width=\"69\" height=\"16\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<div class=\"callout\">\n<h4>Important Tidbits<\/h4>\n<p align=\"CENTER\">Bases of logarithms can be any positive number. The most common base is 10, and<br \/>\nlogarithms with a base 10 are called <abbr title=\"logarithm with base 10 \">common logarithms<\/abbr>. If a base is not identified, it is assumed to be 10.<\/p>\n<h4>Properties of Logarithms<\/h4>\n<p align=\"center\"><strong>Addition Property<\/strong><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p3_clip_image015.gif\" width=\"159\" height=\"14\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"center\"><strong>Subtraction Property<\/strong><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p3_clip_image018.gif\" width=\"160\" height=\"40\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"center\"><strong>Power Property<\/strong><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p3_clip_image021.gif\" width=\"113\" height=\"17\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<h3>How do we change the base of a logarithm?<\/h3>\n<p>When working with logarithms, it is often desirable to convert\u00a0from one base to another. Frequently, we will want to convert to\u00a0base 10. Many calculators only evaluate logarithms for base 10;\u00a0therefore, we have to make the conversion before using the\u00a0calculator to solve. The conversion, or change of base, formula is\u00a0<img decoding=\"async\" class=\"no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image003.gif\" name=\"graphics3\" \/>,\u00a0where <em>b\u00a0<\/em>= the existing\u00a0base,<em> a <\/em>=\u00a0the new base, and <em>q\u00a0<\/em>= a positive\u00a0number.\u00a0Let\u2019s try an example. Convert\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image006.gif\" width=\"35\" height=\"16\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0to base 10 and evaluate.<\/p>\n<p><strong>Step 1:<\/strong> In\u00a0this problem, <em>b <\/em>=\u00a04, <em>a <\/em>=\u00a010, and <em>q <\/em>=\u00a08.<\/p>\n<p><strong>Step 2:<\/strong> Use the change of base formula.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image009.gif\" width=\"208\" height=\"45\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h4>How do we solve exponential equations?<\/h4>\n<p>In order to solve an exponential equation, we must first\u00a0isolate the variable located in the exponent. Once isolated, we\u00a0can solve the equation.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Remember to use the relationship between exponential equations and logarithmic<br \/>\nequations: <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image012.gif\" width=\"205\" height=\"17\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<p>For example, solve\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image015.gif\" width=\"68\" height=\"16\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p><strong>Step 1:<\/strong> Recognize that this is an exponential\u00a0equation, with base 3. Convert to logarithmic form.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image018.gif\" width=\"95\" height=\"16\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 2:<\/strong> Convert from base 3 to base 10.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image021.gif\" width=\"206\" height=\"42\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 3:<\/strong> Solve\u00a0for <em>x<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image024.gif\" width=\"68\" height=\"35\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Let&#8217;s try another example. Solve\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image027.gif\" width=\"100\" height=\"16\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p><strong>Step 1:<\/strong> Recognize that this is a logarithmic\u00a0equation, with base 2. Convert to exponential form.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image030.gif\" width=\"69\" height=\"14\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 2:<\/strong> Solve\u00a0for <em>x<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s9_p4_clip_image033.gif\" width=\"78\" height=\"87\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>Some conic sections, formed by the\u00a0intersection of a single or double cone and a geometric plane,\u00a0are:\n<ul>\n<li>circles,<\/li>\n<li>parabolas, and<\/li>\n<li>ellipses.<\/li>\n<\/ul>\n<\/li>\n<li>Each conic has a standard form\u00a0that determines the shape of the graph.\u00a0The distance from the <em><strong>focus\u00a0point<\/strong><\/em> to the <strong><em>vertex<\/em><\/strong> of a\u00a0parabola is the same as the distance from the <strong><em>directrix\u00a0<\/em><\/strong>to the vertex.<\/li>\n<li><strong><em>Integer <\/em><\/strong>and\u00a0<strong><em>fractional exponents<\/em><\/strong> follow the exact\u00a0same rules for operations.<\/li>\n<li><em><strong>Logarithms\u00a0<\/strong><\/em>and the ability to convert between logarithmic and exponential\u00a0form are essential in solving exponential equations.<\/li>\n<li>The change of base formula helps to convert between\u00a0different bases when working with logarithms.<\/li>\n<\/ul>\n<h3>Further Reading in Algebra &amp; Functions<\/h3>\n<p><em>Algebra<\/em> <em>Demystified: A Self-teaching Guide <\/em>(Rhonda\u00a0Huettenmueller): McGraw-Hill, 2003.<\/p>\n<p><em>College Algebra<\/em> (Ron Larson and Robert P. Hostetler):\u00a0Houghton Mifflin, 2003.<\/p>\n<p><em>Practical Algebra: A Self-Teaching Guide<\/em> (Peter H. Selby\u00a0and Steve Slavin): John Wiley and Sons, 1991.<\/p>\n<p><em>Schaum&#8217;s Outline of Intermediate Algebra<\/em> (Ray Steege and\u00a0Kerry Bailey): McGraw Hill, 1997.<\/p>\n<p align=\"CENTER\"><strong><em>Don&#8217;t forget to test your knowledge\u00a0with the <a href=\"http:\/\/www.abcte.org\/drupal\/courses\/mrc\/quizzes\/algfunct2\" target=\"popsome\"> Algebra and Functions II Chapter Quiz; <\/a><\/em><\/strong><\/p>\n<p>&nbsp;<\/p>\n<\/section>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/laws-of-fractional-exponents\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Next Workshop \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Workshop \u27a1 Exponential Functions Objective In this lesson, you will study the standard form of an exponential function and behavior of growth and decay in\u00a0functions. You will also review how to solve equations in exponential form (the variable in the equation is located\u00a0in the exponent). Previously Covered: Fractional\u00a0exponents obey the same [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-93","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/93","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=93"}],"version-history":[{"count":11,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/93\/revisions"}],"predecessor-version":[{"id":474,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/93\/revisions\/474"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=93"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}