{"id":477,"date":"2024-09-07T10:59:14","date_gmt":"2024-09-07T10:59:14","guid":{"rendered":"https:\/\/www.mathros.net.ua\/en\/?p=477"},"modified":"2025-11-06T11:42:31","modified_gmt":"2025-11-06T11:42:31","slug":"logarithmic-equations-with-same-bases","status":"publish","type":"post","link":"https:\/\/www.mathros.net.ua\/en\/logarithmic-equations-with-same-bases.html","title":{"rendered":"Logarithmic Equations with Same Bases: Your Guide to Solving Them"},"content":{"rendered":"<p>Logarithmic equations may seem complex at first, but once you break them down, they\u2019re not so intimidating. In this article, we\u2019ll explore the key concepts behind logarithmic equations, focusing on those with the same bases. We\u2019ll begin with the theoretical foundation and then move on to practical exercises to solidify your understanding. Ready to dive in? Let\u2019s go!<\/p>\n<h2>Logarithmic Equations: Basics and Definitions<\/h2>\n<p>At its core, a logarithmic equation is simply an equation that includes a logarithm with an unknown variable, usually in the argument or the base. These equations can be made easier to solve by converting them into exponential form. Let\u2019s first recall the definition of a logarithm.<\/p>\n<h3>Definition of a Logarithm<\/h3>\n<p>A logarithm is the inverse of an exponent. If you have an exponential equation in the form:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-10022086 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy1.jpg\" alt=\"exponential equation\" width=\"39\" height=\"13\" \/><\/p>\n<p>You can rewrite this in logarithmic form as:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-10022088 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy2.jpg\" alt=\"logarithmic equation\" width=\"65\" height=\"14\" \/><\/p>\n<p>Where:<\/p>\n<ul>\n<li><em>a<\/em>\u00a0is the base;<\/li>\n<li><em>n<\/em> is the argument;<\/li>\n<li><em>x<\/em> is the exponent.<\/li>\n<\/ul>\n<p>This simple definition lays the groundwork for solving logarithmic equations. If the unknown variable is in the argument, you can rewrite the logarithm as an exponential equation to solve for that variable.<\/p>\n<h2>Logarithmic Equations with the Same Bases? Here&#8217;s How to Solve Them!<\/h2>\n<p>Now that we\u2019ve reviewed the basics, let\u2019s discuss how to tackle logarithmic equations where both sides of the equation have logarithms with the same base.<\/p>\n<h3>Key Properties of Logarithms<\/h3>\n<p>There are three key properties that make solving logarithmic equations easier:<\/p>\n<table>\n<tbody>\n<tr>\n<th>Term<\/th>\n<th>Explanation<\/th>\n<th>Formula<\/th>\n<\/tr>\n<tr>\n<td>Product rule<\/td>\n<td>This is helpful when adding logarithms with the same base<\/td>\n<td style=\"text-align: center;\"><img decoding=\"async\" class=\"size-full wp-image-10022090 aligncenter\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy3.jpg\" alt=\"product rule\" width=\"165\" height=\"14\" \/><\/td>\n<\/tr>\n<tr>\n<td>Quotient rule<\/td>\n<td>This comes in handy when subtracting logarithms<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-10022092 aligncenter\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy4.jpg\" alt=\"quotient rule\" width=\"153\" height=\"28\" \/><\/td>\n<\/tr>\n<tr>\n<td>Power rule<\/td>\n<td>This property is used when the argument of the logarithm is raised to a power<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-10022094 aligncenter\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy5.jpg\" alt=\"power rule\" width=\"119\" height=\"15\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Solving Logarithmic Equations: Practical Examples with Solutions<\/h2>\n<p>Let\u2019s now move on to some exercises that will help reinforce what we\u2019ve learned so far. These examples will show you step-by-step how to apply the properties of logarithms and convert them into exponential form for solving.<\/p>\n<h6>Example 1: Solve the logarithmic equation log<sub>2<\/sub>(x+3)=1<\/h6>\n<p>We convert it to exponential form:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022097 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy6.jpg\" alt=\"exponential equation\" width=\"58\" height=\"14\" \/><\/p>\n<p>Simplify this equation to find <em>x<\/em>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022098 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy7.jpg\" alt=\"solution of the logarithmic equation\" width=\"170\" height=\"11\" \/><\/p>\n<p>So, the solution to the logarithmic equation is <em>x=-1<\/em>.<\/p>\n<h6>Example 2: Solve the logarithmic equation log<sub>x+2<\/sub>(64)=2<\/h6>\n<p>This equation is a bit more complex. Let\u2019s convert it to exponential form:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022100 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy8.jpg\" alt=\"exponential equation\" width=\"75\" height=\"15\" \/><\/p>\n<p>Take the square root of both sides and find <em>x<\/em>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022101 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy9.jpg\" alt=\"solution of the logarithmic equation\" width=\"161\" height=\"11\" \/><\/p>\n<p>Thus, the solution is <em>x=6<\/em>.<\/p>\n<h3>Solving Logarithmic Equations with Multiple Terms<\/h3>\n<p>When an equation has several logarithmic terms, the key is to use the properties of logarithms to simplify the equation. Let\u2019s look at an example of how to solve such equations with the same bases.<\/p>\n<h6>Example 3: Solve the logarithmic equation log<sub>5<\/sub>(x-1)+log<sub>5<\/sub>(3)=log<sub>5<\/sub>(15)<\/h6>\n<p>We see that there is a sum of logarithms with the same base on the left-hand side, so we can use the product law to combine them:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022103 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy10.jpg\" alt=\"logarithmic equation\" width=\"149\" height=\"17\" \/><\/p>\n<p>Simplify the multiplication:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022104 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy11.jpg\" alt=\"logarithmic equation\" width=\"139\" height=\"14\" \/><\/p>\n<p>Since the logarithms have the same base, we can eliminate them and form an equation with the arguments:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022105 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy12.jpg\" alt=\"equation with the arguments\" width=\"75\" height=\"11\" \/><\/p>\n<p>Solve the linear equation:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022111 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy17.jpg\" alt=\"solution of the logarithmic equation\" width=\"234\" height=\"27\" \/><\/p>\n<p>So, the solution is <em>x=4<\/em>.<\/p>\n<h6>Example 4: Solve the logarithmic equation log<sub>3<\/sub>(x+3)-log<sub>3<\/sub>(2)=log<sub>3<\/sub>(x-1)-log<sub>3<\/sub>(7)<\/h6>\n<p>In this case, we have a difference of logarithms on both sides, so we can apply the quotient rule of logarithms:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022108 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy14.jpg\" alt=\"logarithmic equation\" width=\"145\" height=\"27\" \/><\/p>\n<p>Since the expressions inside the logarithms cannot be simplified further, we can eliminate the logarithms and solve the equation with the arguments:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022109 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy15.jpg\" alt=\"equation with the arguments\" width=\"75\" height=\"27\" \/><\/p>\n<p>Now, cross-multiply:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022110 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy16.jpg\" alt=\"linear equation\" width=\"124\" height=\"13\" \/><\/p>\n<p>Expand both sides and solve the linear equation:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022113 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy18.jpg\" alt=\"solution of the logarithmic equation\" width=\"414\" height=\"27\" \/><\/p>\n<p>So, the solution to the logarithmic equation is <em>x=-4.6<\/em>.<\/p>\n<h6>Example 5: Solve the logarithmic equation log(x<sup>2<\/sup>)+0.5\u22c5log(4)=log(x<sup>2<\/sup>+16)<\/h6>\n<p>We notice that the logarithms in this equation do not specify a base. When logarithms do not have a specified base, we assume the base is <em>10<\/em>. Logarithms with base <em>10<\/em> are called <a title=\"What is a common logarithm\" href=\"https:\/\/en.wikipedia.org\/wiki\/Common_logarithm\" target=\"_blank\" rel=\"nofollow noopener\">common logarithms<\/a>.<\/p>\n<p>In this equation, we can start by using the power rule to rewrite the logarithm with a coefficient:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022116 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy19.jpg\" alt=\"logarithmic equation\" width=\"181\" height=\"16\" \/><\/p>\n<p>Since <em>4<sup>0.5<\/sup>=2<\/em>, the equation becomes:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022117 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy20.jpg\" alt=\"logarithmic equation\" width=\"169\" height=\"16\" \/><\/p>\n<p>Next, apply the product rule to the left side:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022119 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy21.jpg\" alt=\"logarithmic equation\" width=\"138\" height=\"16\" \/><\/p>\n<p>Since the logarithms have the same base, we can eliminate them and form an equation with the arguments:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022120 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy22.jpg\" alt=\"equation with the arguments\" width=\"87\" height=\"14\" \/><\/p>\n<p>Solve the quadratic equation:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10022121 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/09\/logaryfmichni-rivnjannja-z-odnakovymy-osnovamy23.jpg\" alt=\"solution of the logarithmic equation\" width=\"261\" height=\"14\" \/><\/p>\n<p>Thus, we have two solutions: <em>x=4<\/em> and <em>x=-4<\/em>.<\/p>\n<h2>Summary: What&#8217;s Next?<\/h2>\n<p>By now, you probably feel more confident in solving logarithmic equations with the same bases. We&#8217;ve covered key logarithmic properties such as the product, quotient, and power rules, and you&#8217;ve seen how to apply them in practice. But this is just a small part of the mathematical world of logarithms!<\/p>\n<p>Are you ready for the next step? If so, you&#8217;ll definitely want to check out the following topic:<\/p>\n<ol style=\"list-style: none;\">\n<li><a title=\"Logarithmic equations with different bases\" href=\"https:\/\/www.mathros.net.ua\/en\/\">Logarithmic Equations with Different Bases: How to Solve Them.<\/a><\/li>\n<\/ol>\n<p>This topic will help you truly harness the power of logarithms and tackle even more complex problems. Don\u2019t miss the opportunity to expand your knowledge!<\/p>\n<p>Remember, practice is your best ally in mathematics. Keep solving problems, experimenting with new approaches, and soon even the most challenging equations will be within your reach. Good luck, and may logarithms become your best friends!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Logarithmic equations may seem complex at first, but once you break them down, they\u2019re not so intimidating. In this article,<\/p>\n","protected":false},"author":1,"featured_media":478,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"template-centered.php","format":"standard","meta":{"footnotes":""},"categories":[93],"tags":[97,98,94,96,95],"class_list":["post-477","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logarithmic-equations","tag-exponential-equations","tag-logarithm-properties","tag-logarithmic-equations","tag-math-problem-solving","tag-same-base-logarithms"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/477","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/comments?post=477"}],"version-history":[{"count":1,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/477\/revisions"}],"predecessor-version":[{"id":479,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/477\/revisions\/479"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/media\/478"}],"wp:attachment":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/media?parent=477"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/categories?post=477"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/tags?post=477"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}