{"id":2615,"date":"2025-08-13T18:14:01","date_gmt":"2025-08-13T18:14:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2615"},"modified":"2026-03-16T15:51:00","modified_gmt":"2026-03-16T15:51:00","slug":"exponential-and-logarithmic-functions-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-functions-background-youll-need-1\/","title":{"raw":"Exponential and Logarithmic Functions: Background You'll Need 1","rendered":"Exponential and Logarithmic Functions: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Simplify expressions with exponents<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Exponential Notation<\/h2>\r\nExponential notation is a compact way to represent repeated multiplication of the same number. It's a fundamental concept in mathematics that has significant implications across various disciplines, including science, finance, and technology.\r\n\r\nIn exponential notation, a number written with an exponent, like [latex]a^m[\/latex], succinctly expresses that the base [latex]a[\/latex] is multiplied by itself [latex]m[\/latex]\u00a0times. This notation is not only a shorthand for extensive multiplication but also a gateway to understanding more complex mathematical concepts like exponential functions and logarithms.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>exponential notation<\/h3>\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"464\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224353\/CNX_BMath_Figure_10_02_013_img.png\" alt=\"On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.\" width=\"464\" height=\"102\" \/> Exponential notation shows repeated multiplication in a compact form[\/caption]\r\n\r\n&nbsp;\r\n\r\nThis is read [latex]a[\/latex] to the [latex]{m}^{\\mathrm{th}}[\/latex] power.\r\n\r\n&nbsp;\r\n\r\nIn the expression [latex]{a}^{m}[\/latex], the exponent tells us how many times we use the base [latex]a[\/latex] as a factor.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]321375[\/ohm_question]<\/section><section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Negatives and exponents<\/h3>\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"75\" height=\"66\" \/>\r\nCaution! Whether to include a negative sign as part of a base or not often leads to confusion. To clarify\u00a0whether a negative sign is applied before or after the exponent, here is an example.\r\n\r\n&nbsp;\r\n\r\nWhat is the difference in the way you would evaluate these two terms?\r\n<ol>\r\n \t<li style=\"text-align: left;\">[latex]-{3}^{2}[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[latex]{\\left(-3\\right)}^{2}[\/latex]<\/li>\r\n<\/ol>\r\nTo evaluate 1), you would apply the exponent to the three first, then apply the negative sign last, like this:\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}-\\left({3}^{2}\\right)\\\\=-\\left(9\\right) = -9\\end{array}[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left;\">To evaluate 2), you would apply the exponent to the 3 and the negative sign:<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{\\left(-3\\right)}^{2}\\\\=\\left(-3\\right)\\cdot\\left(-3\\right)\\\\={ 9}\\end{array}[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left;\">The key to remembering this is to follow the order of operations. The first expression does not include parentheses so you would apply the exponent to the integer [latex]3[\/latex] first, then apply the negative sign. The second expression includes parentheses, so hopefully you will remember that the negative sign also gets squared.<\/p>\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]321376[\/ohm_question]<\/section>\r\n<h2>Evaluate Expressions With Exponents<\/h2>\r\nEvaluating expressions containing exponents is the same as evaluating the linear expressions from earlier in the course. You substitute the value of the variable into the expression and simplify. So, when you evaluate the expression [latex]5x^{3}[\/latex]\u00a0if [latex]x=4[\/latex], first substitute the value [latex]4[\/latex] for the variable [latex]x[\/latex]. Then evaluate, using order of operations.\r\n\r\n<section class=\"textbox example\">Evaluate\u00a0[latex]5x^{3}[\/latex] if [latex]x=4[\/latex].[reveal-answer q=\"411363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"411363\"]Substitute [latex]4[\/latex] for the variable [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]5\\cdot4^{3}[\/latex]<\/p>\r\nEvaluate [latex]4^{3}[\/latex]. Multiply.\r\n<p style=\"text-align: center;\">[latex]5\\left(4\\cdot4\\cdot4\\right)=5\\cdot64=320[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]5x^{3}=320[\/latex]\u00a0when [latex]x=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>In the example below, notice the how adding parentheses can change the outcome when you are simplifying terms with exponents.\r\n\r\n<section class=\"textbox example\">Evaluate [latex]\\left(5x\\right)^{3}[\/latex]\u00a0if [latex]x=4[\/latex].[reveal-answer q=\"362021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"362021\"]Substitute [latex]4[\/latex] for the variable [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(5\\cdot4\\right)^3[\/latex]<\/p>\r\nMultiply inside the parentheses, then apply the exponent\u2014following the rules of PEMDAS.\r\n<p style=\"text-align: center;\">[latex]20^{3}[\/latex]<\/p>\r\nEvaluate [latex]20^{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex]20\\cdot20\\cdot20=8,000[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(5x\\right)^3=8,000[\/latex] when [latex]x=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Simplify expressions with exponents<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Exponential Notation<\/h2>\n<p>Exponential notation is a compact way to represent repeated multiplication of the same number. It&#8217;s a fundamental concept in mathematics that has significant implications across various disciplines, including science, finance, and technology.<\/p>\n<p>In exponential notation, a number written with an exponent, like [latex]a^m[\/latex], succinctly expresses that the base [latex]a[\/latex] is multiplied by itself [latex]m[\/latex]\u00a0times. This notation is not only a shorthand for extensive multiplication but also a gateway to understanding more complex mathematical concepts like exponential functions and logarithms.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>exponential notation<\/h3>\n<p>&nbsp;<\/p>\n<figure style=\"width: 464px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224353\/CNX_BMath_Figure_10_02_013_img.png\" alt=\"On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.\" width=\"464\" height=\"102\" \/><figcaption class=\"wp-caption-text\">Exponential notation shows repeated multiplication in a compact form<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>This is read [latex]a[\/latex] to the [latex]{m}^{\\mathrm{th}}[\/latex] power.<\/p>\n<p>&nbsp;<\/p>\n<p>In the expression [latex]{a}^{m}[\/latex], the exponent tells us how many times we use the base [latex]a[\/latex] as a factor.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm321375\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321375&theme=lumen&iframe_resize_id=ohm321375&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Negatives and exponents<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"75\" height=\"66\" \/><br \/>\nCaution! Whether to include a negative sign as part of a base or not often leads to confusion. To clarify\u00a0whether a negative sign is applied before or after the exponent, here is an example.<\/p>\n<p>&nbsp;<\/p>\n<p>What is the difference in the way you would evaluate these two terms?<\/p>\n<ol>\n<li style=\"text-align: left;\">[latex]-{3}^{2}[\/latex]<\/li>\n<li style=\"text-align: left;\">[latex]{\\left(-3\\right)}^{2}[\/latex]<\/li>\n<\/ol>\n<p>To evaluate 1), you would apply the exponent to the three first, then apply the negative sign last, like this:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}-\\left({3}^{2}\\right)\\\\=-\\left(9\\right) = -9\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">To evaluate 2), you would apply the exponent to the 3 and the negative sign:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{\\left(-3\\right)}^{2}\\\\=\\left(-3\\right)\\cdot\\left(-3\\right)\\\\={ 9}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">The key to remembering this is to follow the order of operations. The first expression does not include parentheses so you would apply the exponent to the integer [latex]3[\/latex] first, then apply the negative sign. The second expression includes parentheses, so hopefully you will remember that the negative sign also gets squared.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm321376\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321376&theme=lumen&iframe_resize_id=ohm321376&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Evaluate Expressions With Exponents<\/h2>\n<p>Evaluating expressions containing exponents is the same as evaluating the linear expressions from earlier in the course. You substitute the value of the variable into the expression and simplify. So, when you evaluate the expression [latex]5x^{3}[\/latex]\u00a0if [latex]x=4[\/latex], first substitute the value [latex]4[\/latex] for the variable [latex]x[\/latex]. Then evaluate, using order of operations.<\/p>\n<section class=\"textbox example\">Evaluate\u00a0[latex]5x^{3}[\/latex] if [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q411363\">Show Solution<\/button><\/p>\n<div id=\"q411363\" class=\"hidden-answer\" style=\"display: none\">Substitute [latex]4[\/latex] for the variable [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]5\\cdot4^{3}[\/latex]<\/p>\n<p>Evaluate [latex]4^{3}[\/latex]. Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]5\\left(4\\cdot4\\cdot4\\right)=5\\cdot64=320[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]5x^{3}=320[\/latex]\u00a0when [latex]x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>In the example below, notice the how adding parentheses can change the outcome when you are simplifying terms with exponents.<\/p>\n<section class=\"textbox example\">Evaluate [latex]\\left(5x\\right)^{3}[\/latex]\u00a0if [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q362021\">Show Solution<\/button><\/p>\n<div id=\"q362021\" class=\"hidden-answer\" style=\"display: none\">Substitute [latex]4[\/latex] for the variable [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(5\\cdot4\\right)^3[\/latex]<\/p>\n<p>Multiply inside the parentheses, then apply the exponent\u2014following the rules of PEMDAS.<\/p>\n<p style=\"text-align: center;\">[latex]20^{3}[\/latex]<\/p>\n<p>Evaluate [latex]20^{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]20\\cdot20\\cdot20=8,000[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(5x\\right)^3=8,000[\/latex] when [latex]x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":105,"module-header":"background_you_need","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2615"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2615\/revisions"}],"predecessor-version":[{"id":5868,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2615\/revisions\/5868"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2615\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2615"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2615"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2615"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2615"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}