Solving 27^x = 9^(x-4) A Step-by-Step Explanation

Jul 11, 2025 by ADMIN 50 views

Solving the Exponential Equation: 27^x = 9^(x-4)

In mathematics, exponential equations play a crucial role in various fields, including calculus, physics, and engineering. Solving exponential equations often involves manipulating the equations using the properties of exponents and logarithms. This article delves into the step-by-step solution of the exponential equation 27^x = 9^(x-4), providing a clear understanding of the underlying concepts and techniques. Through a detailed explanation, we aim to equip readers with the ability to tackle similar problems confidently. Exponential equations are fundamental in modeling real-world phenomena such as population growth, radioactive decay, and compound interest. Therefore, mastering the methods to solve these equations is an invaluable skill for anyone pursuing studies or careers in quantitative disciplines. The structure of this article will guide you from the initial problem statement to the final solution, ensuring a comprehensive grasp of the solution process. We will begin by rewriting the equation using a common base, which is a crucial step in simplifying exponential equations. Then, we will equate the exponents and solve the resulting linear equation. Finally, we will verify our solution to ensure accuracy. By the end of this discussion, you will not only understand how to solve this specific equation but also gain insights into the broader techniques applicable to solving exponential equations.

Understanding Exponential Equations

Before diving into the specific solution, it’s important to grasp the essence of exponential equations. An exponential equation is an equation in which the variable appears in the exponent. These equations often require a different set of techniques compared to algebraic equations where the variable is in the base. The key to solving exponential equations lies in understanding and applying the properties of exponents. These properties allow us to manipulate the equations into a form where we can easily isolate the variable. One of the most fundamental properties is that if a^m = a^n, then m = n, provided that the base 'a' is the same. This property is the cornerstone of solving many exponential equations, and we will utilize it in solving 27^x = 9^(x-4). Another important property is the power of a power rule, which states that (a^m)n = a^(m*n). This rule enables us to simplify expressions where an exponent is raised to another exponent. Additionally, understanding how to express numbers as powers of a common base is crucial. In the given equation, both 27 and 9 can be expressed as powers of 3, which is the key to simplifying the equation. The general approach to solving exponential equations involves rewriting the equation so that both sides have the same base, then equating the exponents and solving the resulting equation. This method transforms the exponential equation into a more manageable algebraic equation. Exponential equations are not just theoretical constructs; they have practical applications in various fields. For example, in finance, they are used to calculate compound interest; in biology, they model population growth; and in physics, they describe radioactive decay. Therefore, understanding how to solve these equations is crucial for many real-world applications.

Step 1: Rewriting with a Common Base

The initial step in solving the equation 27^x = 9^(x-4) is to rewrite both sides with a common base. Recognizing that both 27 and 9 are powers of 3 is crucial here. Specifically, 27 can be expressed as 3^3, and 9 can be expressed as 3^2. This transformation allows us to use the properties of exponents to simplify the equation. By expressing both sides with the same base, we can then equate the exponents, which is a fundamental principle in solving exponential equations. The process of finding a common base involves identifying the prime factors of the numbers involved. In this case, the prime factorization of 27 is 3 × 3 × 3, which is 3^3, and the prime factorization of 9 is 3 × 3, which is 3^2. Once we have identified the common base, we substitute these expressions back into the original equation. This gives us (3³⁾x = (3²⁾(x-4). The next step involves applying the power of a power rule, which states that (a^m)n = a^(m*n). Applying this rule to both sides of the equation, we get 3^(3x) = 3^(2(x-4)). This transformation is a pivotal step as it allows us to move from the exponential form to a linear equation by equating the exponents. The ability to rewrite numbers with a common base is a technique that is frequently used in solving exponential equations. It simplifies the equation and makes it easier to solve. Understanding this concept is essential for tackling more complex exponential problems. This step not only simplifies the current equation but also lays the foundation for solving other exponential equations that may appear in various mathematical and scientific contexts. The choice of the common base is often dictated by the numbers involved in the equation, and practice in identifying these bases is key to mastering the solution of exponential equations.

Step 2: Equating the Exponents

After rewriting the equation 27^x = 9^(x-4) with a common base of 3, we have 3^(3x) = 3^(2(x-4)). The next crucial step is to equate the exponents. This is a valid operation because if a^m = a^n, then m = n. Applying this principle, we can set the exponents equal to each other, resulting in the equation 3x = 2(x-4). This transformation converts the exponential equation into a simple linear equation, which is much easier to solve. Equating the exponents is a powerful technique in solving exponential equations, and it simplifies the problem significantly. Once we have the linear equation, we can use standard algebraic methods to find the value of x. The equation 3x = 2(x-4) is now in a form that we can manipulate using basic algebraic operations. The next step involves distributing the 2 on the right side of the equation, which gives us 3x = 2x - 8. This step is crucial for simplifying the equation further and isolating the variable x. By equating exponents, we have effectively removed the exponential nature of the equation and transformed it into a problem that can be solved using familiar algebraic techniques. This process highlights the importance of understanding the properties of exponents and how they can be used to simplify and solve equations. The ability to transition from an exponential equation to a linear equation is a key skill in mathematics, particularly in algebra and calculus. The validity of equating the exponents rests on the one-to-one property of exponential functions. This property ensures that if two exponential expressions with the same base are equal, their exponents must also be equal. Therefore, equating the exponents is a logical and mathematically sound step in solving exponential equations.

Step 3: Solving the Linear Equation

Having equated the exponents in the equation 27^x = 9^(x-4), we arrived at the linear equation 3x = 2(x - 4). The next step is to solve this linear equation for x. This involves using basic algebraic manipulations to isolate x on one side of the equation. First, distribute the 2 on the right side of the equation: 3x = 2x - 8. Then, subtract 2x from both sides to get the x terms on one side: 3x - 2x = -8. This simplifies to x = -8. Thus, the solution to the linear equation is x = -8. Solving linear equations is a fundamental skill in algebra, and it is essential for solving various types of mathematical problems, including exponential equations. The process involves performing operations on both sides of the equation to maintain equality while isolating the variable. In this case, subtracting 2x from both sides was the key step in isolating x. The solution x = -8 is a critical result, as it provides the value of x that satisfies the original exponential equation. However, it is always a good practice to verify this solution by substituting it back into the original equation to ensure its correctness. The steps involved in solving this linear equation demonstrate the importance of following algebraic rules and principles. Each step must be performed correctly to arrive at the accurate solution. Linear equations are the foundation of many mathematical concepts, and mastering the techniques to solve them is crucial for further studies in mathematics and related fields. The result x = -8 is not only a numerical answer but also represents a specific point on the number line that satisfies the given equation. This illustrates the connection between algebraic solutions and their graphical representations. The process of solving for x in this linear equation showcases the logical and systematic approach required in mathematics. Each step is justified by mathematical principles, ensuring that the solution is both accurate and reliable.

Step 4: Verification

After obtaining the solution x = -8 for the equation 27^x = 9^(x-4), it is crucial to verify the solution. Verification involves substituting the solution back into the original equation to ensure that it holds true. This step is essential to confirm the accuracy of our calculations and the validity of our solution. Substituting x = -8 into the original equation, we have 27^(-8) = 9^(-8-4). This simplifies to 27^(-8) = 9^(-12). To verify this, we can rewrite both sides using a common base of 3. Recall that 27 = 3^3 and 9 = 3^2. Thus, the equation becomes (3³⁾(-8) = (3²⁾(-12). Applying the power of a power rule, we get 3^(-24) = 3^(-24). Since both sides are equal, the solution x = -8 is verified. Verification is an integral part of the problem-solving process in mathematics. It provides confidence in the correctness of the solution and helps identify any errors that may have occurred during the solution process. In this case, verifying the solution confirmed that our steps were accurate and that x = -8 is indeed the solution to the equation. The verification step also highlights the importance of precision in mathematical calculations. A small error in any step can lead to an incorrect solution, and verification helps catch these errors. Moreover, the verification process reinforces the understanding of the properties of exponents and the relationships between different mathematical expressions. By substituting the solution back into the original equation, we are essentially reversing the steps we took to solve the equation, which can provide a deeper understanding of the underlying concepts. The verification step is not just a formality; it is a crucial part of mathematical rigor. It ensures that our solutions are not only mathematically sound but also practically applicable. The equality 3^(-24) = 3^(-24) demonstrates that the solution satisfies the original equation, confirming that x = -8 is the correct solution.

In conclusion, solving the exponential equation 27^x = 9^(x-4) involves several key steps, including rewriting the equation with a common base, equating the exponents, solving the resulting linear equation, and verifying the solution. We began by expressing both 27 and 9 as powers of 3, which allowed us to rewrite the equation as 3^(3x) = 3^(2(x-4)). Equating the exponents led us to the linear equation 3x = 2(x - 4), which we solved to find x = -8. Finally, we verified this solution by substituting it back into the original equation, confirming its correctness. This process illustrates the systematic approach to solving exponential equations, emphasizing the importance of understanding and applying the properties of exponents. Exponential equations are prevalent in various scientific and mathematical contexts, making the ability to solve them a valuable skill. The techniques used in this solution, such as rewriting with a common base and equating exponents, are applicable to a wide range of exponential problems. The verification step underscores the importance of accuracy and precision in mathematical calculations. It ensures that the solution obtained is not only mathematically correct but also satisfies the original problem. Furthermore, this exercise reinforces the connection between different areas of mathematics, such as algebra and exponential functions. The ability to transform an exponential equation into a linear equation demonstrates the power of mathematical manipulation and the importance of mastering fundamental algebraic skills. Overall, solving exponential equations requires a combination of algebraic techniques, understanding of exponential properties, and a systematic approach. The step-by-step solution presented here provides a clear guide for tackling similar problems and enhances the reader's understanding of exponential equations and their applications. The solution x = -8 is a testament to the power of these techniques and the importance of verification in mathematical problem-solving.