Solving 32^(x-1) = 4^(-x+4) A Step-by-Step Guide

In the realm of mathematics, solving exponential equations often presents a fascinating challenge. These equations, where the variable appears in the exponent, require a unique approach compared to typical algebraic equations. This article delves into a specific example, 32^(x-1) = 4^(-x+4), providing a comprehensive, step-by-step solution that not only answers the question but also elucidates the underlying principles of solving such equations.

Understanding Exponential Equations

At the heart of exponential equations lies the concept of exponents, which represent repeated multiplication. When the variable takes on the role of an exponent, the equation's nature shifts, demanding a different set of problem-solving tools. To tackle these equations effectively, a solid understanding of exponential properties and logarithms is indispensable. Exponential equations find applications across various fields, including finance, physics, and computer science, highlighting their practical significance.

The Challenge: Solving 32^(x-1) = 4^(-x+4)

Our main task involves deciphering the equation 32^(x-1) = 4^(-x+4). This equation, at first glance, may seem daunting, but it gracefully surrenders to the application of fundamental mathematical principles. The key lies in recognizing that both 32 and 4 can be expressed as powers of a common base, which is 2. This realization paves the way for simplifying the equation and ultimately solving for the elusive variable, x.

Step 1: Expressing Both Sides with a Common Base

The cornerstone of solving this equation rests on the observation that both 32 and 4 are powers of 2. Specifically, 32 can be written as 2^5, and 4 can be written as 2^2. This transformation is crucial because it allows us to rewrite the equation with a common base, paving the way for simplification. By expressing both sides of the equation using the same base, we can directly compare the exponents and establish a relationship that leads us to the solution. This step exemplifies the elegance of mathematical problem-solving, where seemingly complex problems are reduced to simpler forms through insightful transformations.

Let's rewrite the equation using the common base of 2:

• 32^(x-1) = (2⁵⁾(x-1)
• 4^(-x+4) = (2²⁾(-x+4)

Now our equation looks like this:

(2⁵⁾(x-1) = (2²⁾(-x+4)

Step 2: Applying the Power of a Power Rule

The next pivotal step involves applying the power of a power rule, a fundamental principle in the realm of exponents. This rule elegantly states that when you raise a power to another power, you multiply the exponents. In mathematical notation, this is expressed as (a^m)n = a^(m*n). This rule is not just a mathematical formality; it's a powerful tool that simplifies complex exponential expressions, making them more manageable and revealing the underlying relationships between variables.

Applying this rule to our equation, we multiply the exponents:

• (2⁵⁾(x-1) becomes 2^(5*(x-1)) = 2^(5x-5)
• (2²⁾(-x+4) becomes 2^(2*(-x+4)) = 2^(-2x+8)

Our equation now transforms into a more streamlined form:

2^(5x-5) = 2^(-2x+8)

This simplified equation brings us closer to the solution, as the common base allows us to equate the exponents directly.

Step 3: Equating the Exponents

With the equation now expressed in the form of 2^(5x-5) = 2^(-2x+8), we arrive at a critical juncture where we can leverage the fundamental property of exponential functions: if a^m = a^n, then m = n. This property serves as a bridge, allowing us to transition from the realm of exponential expressions to the more familiar territory of linear equations. By equating the exponents, we effectively strip away the exponential facade, revealing the underlying algebraic relationship that governs the variable, x. This step underscores the power of mathematical equivalencies in simplifying problems and making them more amenable to solution.

Therefore, we can equate the exponents:

5x - 5 = -2x + 8

This equation is a straightforward linear equation, which can be easily solved using basic algebraic techniques.

Step 4: Solving the Linear Equation

We've successfully transformed the original exponential equation into a linear equation: 5x - 5 = -2x + 8. Now, the task at hand is to isolate the variable, x, and determine its value. This involves a series of algebraic manipulations, each guided by the principle of maintaining equality on both sides of the equation. We'll employ techniques such as adding like terms and isolating the variable to one side, gradually unraveling the equation until the value of x is revealed. This process highlights the systematic nature of algebra, where complex equations are methodically broken down into simpler, solvable forms.

Let's solve for x:

1. Add 2x to both sides: 5x + 2x - 5 = -2x + 2x + 8, which simplifies to 7x - 5 = 8
2. Add 5 to both sides: 7x - 5 + 5 = 8 + 5, which simplifies to 7x = 13
3. Divide both sides by 7: (7x)/7 = 13/7, which gives us x = 13/7

Therefore, the solution to the equation is x = 13/7.

Step 5: Verifying the Solution

In the pursuit of mathematical accuracy, verification stands as a crucial step. It's not enough to simply arrive at a solution; we must rigorously test it to ensure its validity. This involves substituting the calculated value of x back into the original equation and confirming that both sides of the equation hold true. Verification acts as a safeguard against errors that may have crept in during the solution process, bolstering our confidence in the correctness of the final answer. This step underscores the importance of precision and attention to detail in mathematical problem-solving, where even a minor error can lead to an incorrect conclusion.

To verify our solution, substitute x = 13/7 into the original equation:

32^(x-1) = 4^(-x+4)

32^((13/7)-1) = 4^(-(13/7)+4)

32^(6/7) = 4^(15/7)

(2⁵⁾(6/7) = (2²⁾(15/7)

2^(30/7) = 2^(30/7)

Since both sides are equal, our solution x = 13/7 is correct.

Conclusion: Mastering Exponential Equations

In conclusion, solving exponential equations like 32^(x-1) = 4^(-x+4) is an exercise in applying fundamental mathematical principles. By expressing both sides with a common base, utilizing the power of a power rule, and equating exponents, we can transform seemingly complex equations into manageable linear equations. The solution, x = 13/7, is a testament to the power of systematic problem-solving and the elegance of mathematical transformations. This journey through exponential equations not only provides a solution but also reinforces the importance of understanding the underlying concepts and techniques that form the bedrock of mathematical proficiency. Mastering these techniques equips us to tackle a wide array of mathematical challenges, enhancing our analytical and problem-solving capabilities.