Solving For X In 3^(4x) = 27^(x-3) A Step By Step Guide

Introduction

In this article, we will delve into the realm of exponential equations, focusing on solving a specific problem: determining the value of x for which the equation 3^(4x) = 27^(x-3) holds true. Exponential equations, characterized by variables appearing in the exponents, are a fundamental concept in algebra and calculus. Mastering the techniques to solve them is crucial for various mathematical applications. Understanding exponential equations is a cornerstone of advanced mathematical studies, and the ability to solve them is vital for success in various fields, including science, engineering, and economics. This article provides a comprehensive guide to tackling such problems, breaking down the steps and underlying principles to ensure clarity and mastery.

Understanding the Basics of Exponential Equations: Before diving into the solution, it's essential to grasp the fundamentals of exponential equations. An exponential equation is one where the variable appears in the exponent. These equations often require manipulating exponents and bases to find the unknown variable. The key to solving these equations lies in the ability to express both sides of the equation with the same base. By doing so, we can equate the exponents and solve for the variable. This principle is based on the fundamental property of exponential functions: if a^m = a^n, then m = n, provided that a is a positive number not equal to 1. This property is the backbone of our approach to solving the given equation. We will leverage this principle to simplify the equation and ultimately find the value of x that satisfies the given condition.

Why Exponential Equations Matter: Exponential equations are not just abstract mathematical concepts; they have profound applications in the real world. They are used to model phenomena such as population growth, radioactive decay, compound interest, and the spread of diseases. The solutions to these equations provide critical insights into the rates and patterns of these processes. For instance, in finance, exponential equations are used to calculate the future value of investments, taking into account the effects of compounding interest over time. In environmental science, they help in predicting the rate of decay of pollutants. In biology, they model the growth of bacterial colonies. Thus, understanding how to solve exponential equations is not only academically important but also practically relevant in numerous fields.

Problem Statement

Our main task is to find the value of x that satisfies the exponential equation:

3^(4x) = 27^(x-3)

This equation presents a classic example of an exponential equation where we need to equate the bases to solve for x. The left side of the equation has a base of 3, while the right side has a base of 27. Recognizing that 27 is a power of 3 (specifically, 3^3) is the crucial first step in simplifying the equation. Once we express both sides with the same base, we can equate the exponents and proceed to solve for x. This problem not only tests our algebraic skills but also our understanding of the properties of exponents and the ability to manipulate exponential expressions. The solution will involve a combination of exponent rules and algebraic techniques, providing a comprehensive exercise in mathematical problem-solving.

Solution

1. Express both sides with the same base:

   • We know that 27 can be written as 3^3. So, we can rewrite the equation as:

     3^(4x) = (3³⁾(x-3)

   This step is critical because it allows us to apply the fundamental property of exponential equations, which states that if the bases are equal, then the exponents must also be equal. By expressing both sides of the equation with the same base, we set the stage for simplifying the equation and isolating the variable x. This transformation is a common strategy in solving exponential equations and is often the key to unlocking the solution. The ability to recognize and apply this principle is a fundamental skill in algebra. This step transforms the original equation into a more manageable form, allowing us to proceed with the next steps in the solution.

2. Simplify the right side using the power of a power rule:

   • The power of a power rule states that (a^m)n = a^(m*n). Applying this rule to the right side, we get:

     3^(4x) = 3^(3*(x-3))

   The power of a power rule is a fundamental concept in algebra and is essential for simplifying exponential expressions. This rule allows us to reduce complex exponential expressions into simpler forms, making it easier to manipulate and solve equations. In this case, applying the rule helps us to eliminate the nested exponent on the right side of the equation, bringing us one step closer to equating the exponents and solving for x. Understanding and applying this rule correctly is crucial for solving exponential equations effectively. This simplification is a key step in our solution process, as it directly leads to a more manageable equation.

3. Equate the exponents:

   • Since the bases are the same (both are 3), we can set the exponents equal to each other:

     4x = 3*(x-3)

   This is the pivotal moment in solving the exponential equation. Once we have expressed both sides with the same base, equating the exponents is a direct application of the fundamental property of exponential functions. This step transforms the exponential equation into a linear equation, which is much easier to solve. This equation sets the stage for the final algebraic manipulations that will lead us to the value of x. The ability to recognize and perform this step is a key skill in solving exponential equations. From here, we can use basic algebraic techniques to isolate x and find the solution.

4. Solve for x:

   • Expand the right side:

     4x = 3x - 9

   • Subtract 3x from both sides:

     4x - 3x = -9

   • Simplify:

     x = -9

   The final steps involve straightforward algebraic manipulations to isolate x and find its value. Expanding the right side of the equation and then subtracting 3x from both sides are standard techniques for solving linear equations. The goal is to get x by itself on one side of the equation, which allows us to directly read off the solution. Each of these steps is crucial to ensure accuracy and lead us to the correct answer. This process demonstrates the importance of mastering basic algebraic skills, which are fundamental for solving a wide range of mathematical problems.

Answer

The value of x that satisfies the equation 3^(4x) = 27^(x-3) is -9.

Therefore, the correct answer is A. -9.

This conclusion is the culmination of our step-by-step solution, where we methodically transformed the exponential equation into a linear equation and solved for x. This final answer confirms that -9 is the value that makes the original equation true. This process underscores the importance of careful algebraic manipulation and a solid understanding of the properties of exponents. The final answer not only provides the solution to the specific problem but also reinforces the techniques and principles used in solving exponential equations.

Conclusion

In this article, we have successfully solved the exponential equation 3^(4x) = 27^(x-3) by employing the principles of equating bases and exponents. The key takeaway is that when solving exponential equations, identifying a common base is crucial. By expressing both sides of the equation with the same base, we can simplify the equation and solve for the unknown variable. This approach is not only effective but also elegant in its simplicity. The ability to recognize and apply this technique is a valuable skill in algebra and beyond. Exponential equations appear in various mathematical and real-world contexts, making the understanding and mastery of their solutions essential for further studies and practical applications.

Recap of the Solution Process: We started by recognizing that 27 is a power of 3, which allowed us to rewrite the equation with a common base. We then applied the power of a power rule to simplify the exponents. By equating the exponents, we transformed the exponential equation into a linear equation, which we solved using basic algebraic techniques. The solution, x = -9, was obtained through a series of logical and methodical steps, each building upon the previous one. This structured approach is a hallmark of effective problem-solving in mathematics. By understanding and practicing these steps, students can confidently tackle similar exponential equations.

Further Exploration of Exponential Equations: Exponential equations are a vast and fascinating area of mathematics. This article provides a foundation for further exploration and study. More complex exponential equations may involve logarithmic functions, which are the inverse of exponential functions. Understanding logarithms is crucial for solving equations where isolating the variable in the exponent is not straightforward. Additionally, exponential functions and equations play a critical role in calculus, particularly in the study of exponential growth and decay. Concepts such as derivatives and integrals are used to analyze the behavior of exponential functions and their applications in various fields. This article serves as a stepping stone towards more advanced topics in mathematics and their real-world applications.