{"id":17255,"date":"2023-05-30T08:57:31","date_gmt":"2023-05-30T05:57:31","guid":{"rendered":"https:\/\/starlanguageblog.com\/?p=17255"},"modified":"2023-05-30T08:57:31","modified_gmt":"2023-05-30T05:57:31","slug":"find-the-solution-to-the-exponential-equation","status":"publish","type":"post","link":"https:\/\/www.starlanguageblog.com\/find-the-solution-to-the-exponential-equation\/","title":{"rendered":"Find The Solution To The Exponential Equation"},"content":{"rendered":"<h1><span style=\"font-weight: 400;\">Find The Solution To The Exponential Equation<\/span><\/h1>\n<p><span style=\"font-weight: 400;\">We must separate the exponentiating variable to solve exponential functions. With &#8220;a&#8221; as the base and &#8220;x&#8221; as the exponent, exponential functions have the general form f(x) = axe. The objective is to identify the value of &#8220;x&#8221; that resolves the inequality or equation. There are various options depending on the precise issue and the desired resolution.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">We can frequently utilize logarithms to remove the variable from the exponent while solving exponential equations. We can simplify the equation and immediately solve for the variable by taking the logarithm of both sides of the equation. The exact challenge at hand, as well as any existing constraints, will determine the logarithm basis to use.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Let&#8217;s use the equation 2 x 16 as an example. By taking the logarithm of both sides and using a logarithm base to solve for &#8220;x,&#8221; we may make the problem simpler. Since the exponential function&#8217;s base is two, we can utilize the logarithm base 2 in this situation. Logarithmically speaking, log2(2x) = log2(16).<\/span><\/p>\n<p><span style=\"font-weight: 400;\">We reduce the equation to x = log2(16) using the logarithm property that says loga(ab) = b. x = 4 is the result of evaluating the right-hand side of the logarithm. Consequently, x = 4 is the equation 2x = 16&#8217;s answer.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The strategy is similar when dealing with inequalities using exponential functions. We can utilize logarithms to remove the variable from the exponent and directly answer the inequality. It&#8217;s crucial to remember that only when the base of the logarithm is positive do logarithms maintain the inequality&#8217;s order.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">It is essential to look for any constraints on the function&#8217;s domain or any potential superfluous solutions that might emerge throughout the solving process while solving exponential functions. Furthermore, the solutions could not be limited to real numbers if the function involved complex numbers.<\/span><\/p>\n<h2><span style=\"font-weight: 400;\">What Are Exponential Equations?<img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-17273\" src=\"https:\/\/wordpress-1162154-4054340.cloudwaysapps.com\/wp-content\/uploads\/2023\/05\/annie-spratt-ORDz1m1-q0I-unsplash-2.webp\" alt=\"What Are Exponential Equations?\" width=\"640\" height=\"480\" \/><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">An exponential equation is an equation with exponents where the exponent (or a part of the exponent) is a variable. For example, 3x = 81, 5x &#8211; 3 = 625, 62y &#8211; 7 = 121, etc. are examples of exponential equations. We may come across the use of exponential equations when we are solving the problems of algebra, compound interest, exponential growth, exponential decay, etc.<\/span><\/p>\n<h3><b>Types Of Exponential Equations<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">There are three types of exponential equations. They are as follows:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Equations with the same bases on both sides (Example: 4x = 42)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Equations with different bases can be made the same. (Example: 4x = 16 which can be written as 4x = 42)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Equations with different bases cannot be made the same. (Example: 4x = 15)<\/span><\/li>\n<\/ul>\n<h2><span style=\"font-weight: 400;\">Equations With Exponents <\/span><span style=\"font-weight: 400;\">We Will Go Into The Subject Of Equations With Exponents, Investigating Their Characteristics, Norms, And Solution Methods<\/span><\/h2>\n<h3><b>Recognizing\u00a0Exponents\u00a0<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">Exponents are a quick way to represent repeated multiplication. They are also referred to as powers or indexes. A base&#8217;s exponent tells us how many times it has been multiplied by itself. For instance, the base and exponent in the expression 23 are 2 and 3, respectively. It denotes a three-fold multiplication of 2 by itself:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">2^3 = 2 \u00d7 2 \u00d7 2 = 8<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Because of exponents&#8217; properties, calculations are made simpler. One important characteristic is the product of powers, which causes the exponents to be added when two integers with the same base but different exponents are multiplied.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">a(m+n) = a(m a)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The power of a power is another characteristic, according to which increasing one exponent by another exponent multiplies the exponents:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(a m ) n = a m n<\/span><\/p>\n<p><span style=\"font-weight: 400;\">An understanding of these properties is necessary for manipulating and resolving equations with exponents.<\/span><\/p>\n<h3><b>Exponentiation Of Equations<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">We must use techniques that decompose the equation and isolate the variable to solve equations with exponents. Here are several methods that are frequently used:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><strong>Simplify both sides:<\/strong> Begin by separating the two sides of the equation. By combining related terms and employing exponent rules, expressions can be made simpler.<\/span><\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><strong>Remove the exponent:<\/strong> Remove the exponent using inverse procedures. Take the square root of both sides to remove the exponent, like in the case where the variable is increased to the power of 2.<\/span><\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Applying logarithms can sometimes help resolve equations involving exponents. Since logarithms are exponentiation&#8217;s inverse operations, they can isolate the variable. Use logarithms with the same base as the exponent on both sides of the equation.<\/span><\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><strong>Factorization:<\/strong> Remove the common base from the equation if any terms have the same base but distinct exponents. You can then solve for the variable and simplify the equation.<\/span><\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><strong>Substitution:<\/strong> In some cases, replacing the exponentiated variable with a different one can simplify the equation. By using this technique, a new variable is added to represent the original variable multiplied by a particular power.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">You can solve for the variable and locate the solution to equations with exponents by using these methods and carefully altering the equation.<\/span><\/p>\n<h2><span style=\"font-weight: 400;\">Exponential Equations Formulas<img decoding=\"async\" class=\"alignnone size-full wp-image-17276\" src=\"https:\/\/wordpress-1162154-4054340.cloudwaysapps.com\/wp-content\/uploads\/2023\/05\/antoine-dautry-05A-kdOH6Hw-unsplash.webp\" alt=\"Exponential Equations Formulas\" width=\"640\" height=\"427\" srcset=\"https:\/\/www.starlanguageblog.com\/wp-content\/uploads\/2023\/05\/antoine-dautry-05A-kdOH6Hw-unsplash.webp 640w, https:\/\/www.starlanguageblog.com\/wp-content\/uploads\/2023\/05\/antoine-dautry-05A-kdOH6Hw-unsplash-300x200.webp 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" \/><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">While solving an exponential equation, the bases on both sides may be the same or not. Here are the formulas used in each case, which we will learn in detail in the upcoming sections.<\/span><\/p>\n<h3><b>Property of Equality for Exponential Equations<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">This property is useful for solving an exponential equation with the same bases. It says that when the bases on both sides of an exponential equation are equal, the exponents must also be equal. i.e.,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">ax = ay\u00a0 x = y.<\/span><\/p>\n<h3><b>Exponential Equations To Logarithmic Form<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">We know that logarithms are nothing but exponents and vice versa. Hence, an exponential equation can be converted into a logarithmic function. This helps in the process of solving an exponential equation with different bases. Here is the formula to convert exponential equations into logarithmic equations.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">bx = a \u21d4 logba = x<\/span><\/p>\n<h2>Solving Exponential Equations With Same Bases<img decoding=\"async\" class=\"alignnone size-full wp-image-17277\" src=\"https:\/\/wordpress-1162154-4054340.cloudwaysapps.com\/wp-content\/uploads\/2023\/05\/dan-cristian-padure-xJLN32FO7AY-unsplash-2-2.webp\" alt=\"Solving Exponential Equations With Same Bases\" width=\"640\" height=\"477\" \/><\/h2>\n<p><span style=\"font-weight: 400;\">Sometimes, an exponential <a href=\"https:\/\/starlanguageblog.com\/the-hydrogen-oxygen-equation\/\">equation<\/a> may have the same base on both sides of the equation. For example, 5 x 53 has the same base five on both sides. Sometimes, though the exponents on both sides are not the same, they can be made the same. For example, 5x = 125. Though it doesn&#8217;t have the same bases on both sides of the equation, they can be made the same by writing it as 5x = 53 (as 125 = 53). To solve the exponential equations in each of these cases, we just apply the property of equality to the exponential equations. We set the exponents to be the same and solved for the variable.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Here is another example where the bases are not the same but can be made to be the same.<\/span><\/p>\n<h3><b>Example: Solve the exponential equation 7y + 1 = 343y<\/b><\/h3>\n<h3><b>Solution<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">We know that 343 is 73. Using this, the given equation can be written as,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">7y + 1 = (73)y<\/span><\/p>\n<p><span style=\"font-weight: 400;\">7y + 1 = 73y<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Now the bases on both sides are the same. So we can set the exponents to be the same.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">y + 1 = 3y<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Subtracting y from both sides,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">2y = 1<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Dividing both sides by 2,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">y = \u00bd<\/span><\/p>\n<h3><b>Example<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">Solve the equation 3x = 81.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">In this example, we have the base three raised to the power of the variable x, and the equation is set equal to 81. Our goal is to determine the value of x that satisfies the equation.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">To solve this equation, we can use the property of logarithms, which states that if ax = b, then loga(b) = x. Applying this property, we can take the logarithm of both sides of the equation with the base 3:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">log3(3x) = log3(81).<\/span><\/p>\n<p><span style=\"font-weight: 400;\">By using the property mentioned above, the equation simplifies to:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">x = log\u2083(81).<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Now, we need to evaluate the logarithm of 81 with base 3. The logarithm represents the exponent to which the base must be raised to obtain the argument. In this case, we want to find the exponent to which three must be raised to obtain 81. Evaluating this logarithm yields:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">x = log\u2083(81) = 4.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Therefore, the solution to the equation 3x = 81 is x = 4. By substituting x = 4 back into the original equation, we can verify that 34 does indeed equal 81.<\/span><\/p>\n<h2><span style=\"font-weight: 400;\">Solving Exponential Equations With Different Bases<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-17275\" src=\"https:\/\/wordpress-1162154-4054340.cloudwaysapps.com\/wp-content\/uploads\/2023\/05\/greg-rosenke-DJWvzY4pkGU-unsplash-3.webp\" alt=\"Solving Exponential Equations With Different Bases\" width=\"640\" height=\"427\" \/><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">Sometimes, the bases on both sides of an exponential equation may not be the same or\u00a0 cannot be made the same. We solve exponential equations using logarithms when the bases are not the same on both sides of the equation. For example, 5x = 3, which neither has the same bases on both sides nor can the bases be made the same. In such cases, we can do one of the following things:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Convert the exponential equation into the logarithmic form using the formula bx = a \u21d4 log ba = x and solve for the variable.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Apply logarithm (log) on both sides of the equation and solve for the variable. In this case, we must use a logarithm property, log am = m log a.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">We will solve the equation 5x = 3 in each of these methods.<\/span><\/p>\n<h3><b>Method 1<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">We will convert 5 x 3 into logarithmic form. Then we get,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">log53 = x<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Using the change of base property,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">x = (log 3) \/ (log 5)<\/span><\/p>\n<h3><b>Method 2<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">We will apply the log on both sides of 5x = 3. Then we get log 5 x log 3. Using the property log am = m log an on the left side of the equation, we get x log 5 = log 3. Dividing both sides by log 5,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">x = (log 3) \/ (log 5)<\/span><\/p>\n<h3><b>Important Notes on Exponential Equations<\/b><\/h3>\n<p><span style=\"font-weight: 400;\">Here are some important notes concerning the exponential equations:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">To solve the exponential equations of the same bases, just set the exponents equal.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Applying logarithms on both sides to solve the exponential equations of different bases.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The exponential equations with the same bases also can be solved using logarithms.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">If an exponential equation has one on any one side, then we can write it as 1 = a0 for any &#8216;a. For example, to solve 5x = 1, we can write it as 5x = 50, then we get x = 0.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">To solve an exponential equation using logarithms, we can apply &#8220;log&#8221; or &#8220;ln&#8221; on both sides.<\/span><\/li>\n<\/ul>\n<h2>FAQ&#8217;s<\/h2>\n<h3>What is an exponential equation?<\/h3>\n<p>An exponential equation is a mathematical equation in which one or both sides contain variables in exponents, such as &#8220;x&#8221; raised to a power.<\/p>\n<h3>How do I find the solution to an exponential equation?<\/h3>\n<p>To find the solution to an exponential equation, you need to isolate the variable (usually denoted as &#8220;x&#8221;) by using logarithms. Take the logarithm of both sides of the equation and solve for &#8220;x&#8221; using algebraic techniques.<\/p>\n<h3>What is a logarithm?<\/h3>\n<p>A logarithm is the inverse operation of exponentiation. It helps to find the exponent to which a specified base must be raised to obtain a given number. In other words, it allows us to solve for the unknown exponent in an exponential equation.<\/p>\n<h3>Are there any restrictions when solving exponential equations?<\/h3>\n<p>Yes, there can be restrictions when solving exponential equations. For example, if the base of the exponential expression is negative or equal to 1, there may be restrictions on the values of &#8220;x&#8221; that satisfy the equation. It&#8217;s important to check for such restrictions while solving.<\/p>\n<h3>Can exponential equations have multiple solutions?<\/h3>\n<p>Yes, exponential equations can have multiple solutions. In some cases, there may be more than one value of &#8220;x&#8221; that satisfies the equation. It&#8217;s essential to check all possible solutions and verify them by substituting them back into the original equation.<\/p>\n<h3>Are there any alternative methods to solve exponential equations?<\/h3>\n<p>Yes, there are alternative methods to solve exponential equations, depending on the specific equation. For instance, if the exponential expression can be written in a different form, such as a quadratic equation, you can use appropriate techniques like factoring or the quadratic formula. However, logarithmic methods are generally the most commonly used and applicable approach for solving exponential equations.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find The Solution To The Exponential Equation We must separate the exponentiating variable to solve exponential functions. With &#8220;a&#8221; as the base and &#8220;x&#8221; as the exponent, exponential functions have the general form f(x) = axe. The objective is to identify the value of &#8220;x&#8221; that resolves the inequality or equation. There are various options [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":17274,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2078],"tags":[2077],"class_list":["post-17255","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths","tag-find-the-solution-to-the-exponential-equation"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.8 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Find The Solution To The Exponential Equation<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.starlanguageblog.com\/find-the-solution-to-the-exponential-equation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Find The Solution To The Exponential Equation\" \/>\n<meta property=\"og:description\" content=\"Find The Solution To The Exponential Equation We must separate the exponentiating variable to solve exponential functions. 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