{"id":2631,"date":"2025-08-13T18:18:52","date_gmt":"2025-08-13T18:18:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2631"},"modified":"2025-09-10T21:30:25","modified_gmt":"2025-09-10T21:30:25","slug":"exponential-and-logarithmic-equations-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-equations-background-youll-need-1\/","title":{"raw":"Exponential and Logarithmic Equations: Background You'll Need 1","rendered":"Exponential and Logarithmic Equations: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Apply rules for integer exponents<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Product Rule for Exponents<\/h2>\r\nThe Product Rule for Exponents is one of the essential rules in algebra that simplifies the process of working with powers. This rule is pivotal when dealing with exponential expressions, particularly when multiplying them. In essence, it tells us that when we multiply two exponents with the same base, we can simply add the exponents to get the new power of the base.\r\n\r\nThis rule is extremely useful in various mathematical and real-world applications, such as calculating compound interest, understanding scientific notation, or solving problems in physics and engineering. By using the Product Rule, we can manage and simplify complex expressions without the need for lengthy multiplication.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>the product rule for exponents<\/h3>\r\nFor any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex],\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].\r\n\r\nTo multiply exponential terms with the same base, add the exponents.\r\n\r\n<\/div>\r\n<\/section><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=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says \"For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex].\"\r\n\r\n<section class=\"textbox example\">Simplify the following:\r\n<p style=\"text-align: center;\">[latex](a^{3})(a^{7})[\/latex]<\/p>\r\n[reveal-answer q=\"356596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"356596\"]The base of both exponents is [latex]a[\/latex], so the product rule applies.\r\n<p style=\"text-align: center;\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n<p style=\"text-align: center;\">[latex]a^{3+7}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<center>Answer: [latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section>When multiplying more complicated terms, multiply the coefficients and then multiply the variables.\r\n\r\n<section class=\"textbox example\">Simplify the following:\r\n<p style=\"text-align: center;\">[latex]5a^{4}\\cdot7a^{6}[\/latex]<\/p>\r\n[reveal-answer q=\"215459\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"215459\"]Multiply the coefficients.\r\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4}\\cdot{a}^{6}[\/latex]<\/p>\r\nThe base of both exponents is [latex]a[\/latex], so the product rule applies. Add the exponents.\r\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4+6}[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{10}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<center>Answer: [latex]5a^{4}\\cdot7a^{6}=35a^{10}[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6797[\/ohm2_question]<\/section>\r\n<h2>The Quotient (Division) Rule for Exponents<\/h2>\r\nThe Quotient Rule for Exponents is as crucial as the Product Rule and serves as its counterpart for division. This rule assists in simplifying expressions where we have exponential terms with the same base being divided. It states that when you divide exponents with the same base, you can subtract the exponents.\r\n\r\nThis rule has significant practical applications, especially in fields that involve calculations of rates of change, decay, or growth when they are decreasing, such as in the case of depreciation in finance or radioactive decay in physics.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>the quotient (division) rule for exponents<\/h3>\r\nFor any non-zero number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex]:\r\n\r\n<center>[latex] \\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/center>\r\nTo divide exponential terms with the same base, subtract the exponents.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\">Evaluate the following:<center>[latex] \\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex]<\/center>[reveal-answer q=\"96156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96156\"]These two exponents have the same base, [latex]4[\/latex]. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{4}^{9-4}}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<center>[latex] \\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}=4^{5}[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section>When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.\r\n\r\n<section class=\"textbox example\">Simplify the following:<center>[latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]<\/center>[reveal-answer q=\"23604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23604\"]Separate into numerical and variable factors.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\r\nSince the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<center>[latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]=[latex] \\displaystyle 6{{x}^{3}}[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6800[\/ohm2_question]<\/section><section>\r\n<h3>The Power Rule for Exponents<\/h3>\r\nAnother word for an exponent is power. You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as [latex]3[\/latex] raised to the [latex]5[\/latex]th power. In this section, we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, what to do when two numbers or variables are multiplied and both are raised to an exponent, and what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>the power rule for exponents<\/h3>\r\nFor any positive number [latex]x[\/latex] and integers [latex]a[\/latex] and [latex]b[\/latex]: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6821[\/ohm2_question]<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Apply rules for integer exponents<\/li>\n<\/ul>\n<\/section>\n<h2>The Product Rule for Exponents<\/h2>\n<p>The Product Rule for Exponents is one of the essential rules in algebra that simplifies the process of working with powers. This rule is pivotal when dealing with exponential expressions, particularly when multiplying them. In essence, it tells us that when we multiply two exponents with the same base, we can simply add the exponents to get the new power of the base.<\/p>\n<p>This rule is extremely useful in various mathematical and real-world applications, such as calculating compound interest, understanding scientific notation, or solving problems in physics and engineering. By using the Product Rule, we can manage and simplify complex expressions without the need for lengthy multiplication.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the product rule for exponents<\/h3>\n<p>For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex],\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/p>\n<p>To multiply exponential terms with the same base, add the exponents.<\/p>\n<\/div>\n<\/section>\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=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says &#8220;For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex].&#8221;<\/p>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<p style=\"text-align: center;\">[latex](a^{3})(a^{7})[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q356596\">Show Solution<\/button><\/p>\n<div id=\"q356596\" class=\"hidden-answer\" style=\"display: none\">The base of both exponents is [latex]a[\/latex], so the product rule applies.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]a^{3+7}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>When multiplying more complicated terms, multiply the coefficients and then multiply the variables.<\/p>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<p style=\"text-align: center;\">[latex]5a^{4}\\cdot7a^{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q215459\">Show Solution<\/button><\/p>\n<div id=\"q215459\" class=\"hidden-answer\" style=\"display: none\">Multiply the coefficients.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4}\\cdot{a}^{6}[\/latex]<\/p>\n<p>The base of both exponents is [latex]a[\/latex], so the product rule applies. Add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4+6}[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{10}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]5a^{4}\\cdot7a^{6}=35a^{10}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6797\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6797&theme=lumen&iframe_resize_id=ohm6797&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>The Quotient (Division) Rule for Exponents<\/h2>\n<p>The Quotient Rule for Exponents is as crucial as the Product Rule and serves as its counterpart for division. This rule assists in simplifying expressions where we have exponential terms with the same base being divided. It states that when you divide exponents with the same base, you can subtract the exponents.<\/p>\n<p>This rule has significant practical applications, especially in fields that involve calculations of rates of change, decay, or growth when they are decreasing, such as in the case of depreciation in finance or radioactive decay in physics.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the quotient (division) rule for exponents<\/h3>\n<p>For any non-zero number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/div>\n<p>To divide exponential terms with the same base, subtract the exponents.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Evaluate the following:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q96156\">Show Solution<\/button><\/p>\n<div id=\"q96156\" class=\"hidden-answer\" style=\"display: none\">These two exponents have the same base, [latex]4[\/latex]. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{4}^{9-4}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}=4^{5}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.<\/p>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q23604\">Show Solution<\/button><\/p>\n<div id=\"q23604\" class=\"hidden-answer\" style=\"display: none\">Separate into numerical and variable factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\n<p>Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]=[latex]\\displaystyle 6{{x}^{3}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6800\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6800&theme=lumen&iframe_resize_id=ohm6800&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h3>The Power Rule for Exponents<\/h3>\n<p>Another word for an exponent is power. You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as [latex]3[\/latex] raised to the [latex]5[\/latex]th power. In this section, we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, what to do when two numbers or variables are multiplied and both are raised to an exponent, and what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the power rule for exponents<\/h3>\n<p>For any positive number [latex]x[\/latex] and integers [latex]a[\/latex] and [latex]b[\/latex]: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6821\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6821&theme=lumen&iframe_resize_id=ohm6821&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\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":510,"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\/2631"}],"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":2,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2631\/revisions"}],"predecessor-version":[{"id":3844,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2631\/revisions\/3844"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/510"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2631\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2631"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2631"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2631"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}