Solving Exponential Equations A Step-by-Step Guide To 81^(x-4) = 27^(-x+4)

Jul 14, 2025 by ADMIN 75 views

Solving Elementary Exponential Equations: A Comprehensive Guide to 81^(x-4) = 27^(-x+4)

Introduction

In the realm of mathematics, exponential equations form a crucial part of algebra. These equations involve variables in the exponents, and solving them often requires a deep understanding of exponential properties and logarithmic functions. In this comprehensive guide, we will dive deep into solving the elementary exponential equation 81^(x-4) = 27^(-x+4). This equation is a classic example of how to manipulate exponential expressions to find unknown variables. Through this detailed exploration, we aim to enhance your problem-solving skills and provide a solid foundation for tackling more complex mathematical challenges. We will begin by breaking down the fundamental concepts and gradually proceed to apply the necessary techniques to solve the equation step-by-step. By the end of this guide, you will not only understand the solution to this specific problem but also gain a broader perspective on how to approach similar exponential equations with confidence and precision. Exponential equations are not just abstract mathematical constructs; they have practical applications in various fields, including finance, science, and engineering. Understanding how to solve them is a valuable skill that can enhance your analytical capabilities and problem-solving proficiency.

Understanding Exponential Equations

Before diving into the solution, it's essential to understand what exponential equations are and the basic principles governing them. Exponential equations are equations in which the variable appears in the exponent. They are of the general form a^x = b, where a is the base, x is the exponent, and b is the result. To solve exponential equations, we often use properties of exponents and logarithms to isolate the variable. One of the key strategies is to rewrite the equation so that both sides have the same base. This allows us to equate the exponents and solve for the variable. Another common technique involves the use of logarithms, which are the inverse functions of exponentials. Logarithms provide a powerful tool for simplifying exponential expressions and solving for unknown exponents. Understanding the properties of exponents is crucial for manipulating exponential equations effectively. For instance, the property a^(m+n) = a^m * a^n allows us to combine exponential terms with the same base. Similarly, the property (a^m)n = a^(mn)* enables us to simplify exponents raised to another power. These properties, along with the properties of logarithms, form the foundation for solving a wide range of exponential equations. In addition to these algebraic techniques, it's important to develop a strong intuition for how exponential functions behave. Exponential functions exhibit rapid growth or decay, depending on the base. Understanding this behavior can help you anticipate the solutions to exponential equations and check the validity of your results. With a solid grasp of these fundamental concepts, you'll be well-equipped to tackle the specific equation at hand and other exponential equations you encounter in your mathematical journey.

Rewriting the Equation with a Common Base

The first step in solving the equation 81^(x-4) = 27^(-x+4) is to express both sides with a common base. This technique simplifies the equation by allowing us to equate the exponents directly. We observe that both 81 and 27 are powers of 3. Specifically, 81 can be written as 3^4, and 27 can be written as 3^3. Rewriting the equation using the base 3, we get: (3⁴⁾(x-4) = (3³⁾(-x+4). Now, we apply the power of a power rule, which states that (a^m)n = a^(m*n). This rule is crucial for simplifying exponential expressions and is a cornerstone of solving exponential equations. Applying this rule to both sides of the equation, we have: 3^(4(x-4)) = 3^(3(-x+4)). This step is pivotal because it transforms the equation into a form where the bases are the same, which means we can now focus on the exponents. The equation is now much simpler to handle, as it allows us to set the exponents equal to each other. This simplification is a common strategy in solving exponential equations, and mastering it will significantly improve your ability to tackle similar problems. The key insight here is recognizing the common base. By expressing both sides of the equation with the same base, we eliminate the exponential component and reduce the problem to a linear equation. This approach is not only effective but also elegant, showcasing the power of mathematical manipulation in simplifying complex problems. With the equation now in a more manageable form, we are ready to proceed to the next step: equating the exponents and solving for the variable x. This step will bring us closer to the solution and demonstrate the practical application of exponential properties.

Equating the Exponents

Now that we have the equation in the form 3^(4(x-4)) = 3^(3(-x+4)), the next step is to equate the exponents. When the bases are the same, the exponents must be equal for the equation to hold true. This principle is fundamental to solving exponential equations and allows us to transition from an exponential equation to a simpler algebraic equation. By equating the exponents, we get: 4(x - 4) = 3(-x + 4). This equation is now a linear equation in terms of x, which is much easier to solve than the original exponential equation. The transition from exponential to linear form is a key technique in solving these types of problems. It simplifies the problem significantly and makes it accessible to standard algebraic methods. The next step involves expanding both sides of the equation to remove the parentheses. This is a straightforward application of the distributive property, a basic but essential tool in algebra. By expanding the expressions, we can further simplify the equation and isolate the variable x. Remember, the goal is to manipulate the equation in a way that brings x to one side and the constants to the other. This process of simplification is a common theme in mathematical problem-solving, and it's important to develop a systematic approach to ensure accuracy. With the exponents equated and the equation in a linear form, we are now well-positioned to proceed to the next step: simplifying and solving for x. This will involve basic algebraic manipulations and will ultimately lead us to the solution of the original exponential equation. The ability to equate exponents and convert an exponential equation into a linear one is a powerful skill that will serve you well in various mathematical contexts.

Solving for x

After equating the exponents, we have the linear equation 4(x - 4) = 3(-x + 4). To solve for x, we first expand both sides of the equation. Applying the distributive property, we get: 4x - 16 = -3x + 12. The next step is to gather the terms involving x on one side of the equation and the constants on the other side. This process involves adding 3x to both sides and adding 16 to both sides. Performing these operations, we obtain: 4x + 3x = 12 + 16. Combining like terms, we simplify the equation to: 7x = 28. Now, to isolate x, we divide both sides of the equation by 7: x = 28 / 7. Finally, performing the division, we find: x = 4. Thus, the solution to the equation is x = 4. It is always a good practice to check our solution by substituting it back into the original equation. This ensures that our answer is correct and that we haven't made any errors in our calculations. Substituting x = 4 into the original equation 81^(x-4) = 27^(-x+4), we get: 81^(4-4) = 27^(-4+4). This simplifies to: 81^0 = 27^0. Since any non-zero number raised to the power of 0 is 1, we have: 1 = 1. This confirms that our solution x = 4 is correct. The process of solving for x involves a series of algebraic manipulations, each building upon the previous step. From expanding the equation to isolating the variable, each operation is performed with the goal of simplifying the equation and bringing us closer to the solution. This systematic approach is a hallmark of effective problem-solving in mathematics. With the solution verified, we have successfully solved the exponential equation. This process demonstrates the power of algebraic techniques in tackling mathematical challenges.

Verification of the Solution

To ensure the accuracy of our solution, it is crucial to verify the solution. We found that x = 4 is the solution to the equation 81^(x-4) = 27^(-x+4). To verify this, we substitute x = 4 back into the original equation and check if both sides are equal. Substituting x = 4 into the equation, we get: 81^(4-4) = 27^(-4+4). Simplifying the exponents, we have: 81^0 = 27^0. Now, recall that any non-zero number raised to the power of 0 is equal to 1. Therefore, 81^0 = 1 and 27^0 = 1. Thus, the equation becomes: 1 = 1. Since both sides of the equation are equal, our solution x = 4 is verified. This verification step is an essential part of the problem-solving process. It provides confidence in the correctness of the solution and helps to catch any potential errors made during the solving process. Verifying the solution is not just a formality; it is a crucial step that ensures the validity of our answer. It also reinforces our understanding of the mathematical principles involved. In this case, the verification process highlights the property that any non-zero number raised to the power of 0 is 1. This property is fundamental to the behavior of exponential functions and is important to remember when solving exponential equations. Furthermore, the verification process demonstrates the importance of careful and systematic work. By substituting the solution back into the original equation, we are essentially retracing our steps and ensuring that each step was performed correctly. This methodical approach is a key to success in mathematics and in problem-solving in general. With the solution verified, we can confidently conclude that x = 4 is indeed the correct solution to the equation 81^(x-4) = 27^(-x+4). This completes our comprehensive guide to solving this elementary exponential equation.

Conclusion

In summary, we have successfully solved the elementary exponential equation 81^(x-4) = 27^(-x+4). The key steps involved were: identifying a common base, rewriting the equation with the common base, equating the exponents, solving the resulting linear equation, and verifying the solution. The solution we found is x = 4, which satisfies the original equation. This process highlights the importance of understanding the properties of exponents and logarithms, as well as the techniques for manipulating algebraic expressions. Exponential equations are a fundamental topic in mathematics, with applications in various fields such as finance, science, and engineering. Mastering the techniques for solving these equations is essential for building a strong foundation in mathematics. The strategy of rewriting exponential equations with a common base is a powerful tool that simplifies the problem and allows us to solve for the unknown variable. This approach is applicable to a wide range of exponential equations and should be a part of every student's problem-solving toolkit. Moreover, the importance of verifying the solution cannot be overstated. This step ensures the accuracy of our answer and helps to catch any potential errors. In addition to the specific techniques used to solve this equation, the general approach of breaking down a problem into smaller, manageable steps is a valuable skill that can be applied to many areas of life. By systematically analyzing the problem, identifying the key principles involved, and applying the appropriate techniques, we can tackle even complex mathematical challenges. We hope that this comprehensive guide has provided you with a deeper understanding of how to solve exponential equations and has enhanced your problem-solving abilities. With practice and perseverance, you can master these techniques and confidently approach similar problems in the future. Remember, mathematics is a journey, and each problem solved is a step forward on that journey.