Solving Exponential Equations: Find X In 27^(-x-1) = 81

Nov 27, 2025 by Alex Johnson 56 views

Are you ready to dive into the world of exponential equations? In this comprehensive guide, we'll break down the steps to solve the equation 27^(-x-1) = 81. Exponential equations might seem daunting at first, but with a systematic approach and a bit of practice, you'll be solving them like a pro. This article aims to provide you with a clear, step-by-step solution, ensuring you understand not just the how, but also the why behind each step. So, let’s get started and unravel this mathematical puzzle together!

Understanding Exponential Equations

Before we jump into solving, let’s quickly recap what exponential equations are and why they matter. In essence, an exponential equation is one where the variable appears in the exponent. These equations are crucial in various fields, from finance (calculating compound interest) to science (modeling population growth and radioactive decay). They're a fundamental part of mathematical literacy and problem-solving.

In this particular case, we have 27^(-x-1) = 81. Our mission is to find the value of 'x' that makes this equation true. To do that effectively, we'll use the properties of exponents and logarithms. The key here is to recognize that both 27 and 81 can be expressed as powers of the same base, which will simplify our equation significantly. Stay tuned as we break down each step in detail!

The Core Principle: Common Bases

The golden rule when solving exponential equations is to try and express all terms with a common base. This allows us to equate the exponents directly, which simplifies the equation into a more manageable form. Why does this work? It's because the exponential function is one-to-one, meaning that if a^m = a^n, then m = n. This property is our key to unlocking the solution.

For the equation 27^(-x-1) = 81, we can express both 27 and 81 as powers of 3. This is the crucial first step. 27 is 3 cubed (3^3), and 81 is 3 to the fourth power (3^4). Recognizing these common bases is essential for simplifying the equation and making it solvable. Once we rewrite the equation using the common base, the path to finding 'x' becomes much clearer. Let’s move on to the next step and see how this transformation helps us.

Step-by-Step Solution

Now, let's walk through the solution step-by-step, making sure each action is clear and easy to follow.

Step 1: Express Both Sides with a Common Base

The first and most crucial step is to rewrite both 27 and 81 as powers of 3. We know that:

27 = 3^3
81 = 3^4

So, we can rewrite our original equation, 27^(-x-1) = 81, as:

(3³⁾(-x-1) = 3^4

This transformation is significant because it sets the stage for using the properties of exponents to simplify the equation further. By having a common base, we can directly compare the exponents, which is the key to solving for 'x'. This step is not just about rewriting the numbers; it’s about changing the perspective and recognizing the underlying structure of the equation. The next step will involve applying the power of a power rule, which will bring us even closer to isolating 'x'.

Step 2: Apply the Power of a Power Rule

The power of a power rule states that (a^m)n = a^(m*n). In our case, we have (3³⁾(-x-1). Applying this rule, we multiply the exponents:

3^(3*(-x-1)) = 3^4

This simplifies to:

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

Applying this rule is a critical step in simplifying exponential equations. It allows us to combine exponents and move closer to a form where we can directly solve for the variable. Without this rule, we would be stuck with a more complex expression that’s difficult to handle. Now that we have a single exponent on each side of the equation, we’re in an excellent position to equate the exponents in the next step. This will transform the exponential equation into a simple linear equation, which is much easier to solve.

Step 3: Equate the Exponents

Since the bases are now the same (both are 3), we can equate the exponents. This is based on the principle that if a^m = a^n, then m = n. So, we have:

-3x - 3 = 4

This step is the heart of solving exponential equations once you’ve established a common base. By equating the exponents, we transform the problem from dealing with exponents to dealing with a simple algebraic equation. This is a significant simplification and brings us much closer to the solution. The beauty of this step lies in its directness and clarity. It bridges the gap between exponential expressions and linear equations, making the problem accessible. In the next step, we’ll solve this linear equation to find the value of 'x'.

Step 4: Solve the Linear Equation

Now we have a simple linear equation: -3x - 3 = 4. Let's solve for 'x'.

First, add 3 to both sides:

-3x = 7

Then, divide both sides by -3:

x = -7/3

Therefore, the solution to the equation 27^(-x-1) = 81 is x = -7/3. Solving this linear equation is a straightforward process, but it’s crucial to perform each step accurately. A mistake in the arithmetic here can lead to an incorrect final answer. This step demonstrates the power of algebraic manipulation in problem-solving. By applying basic operations, we isolate 'x' and reveal its value. This result is the culmination of all the previous steps, confirming that we have successfully navigated the complexities of the exponential equation.

Verification

To ensure our solution is correct, we should always verify it by plugging it back into the original equation.

Step 5: Verify the Solution

Let’s substitute x = -7/3 back into the original equation 27^(-x-1) = 81:

27^(-(-7/3)-1) = 81

First, simplify the exponent:

-(-7/3) - 1 = 7/3 - 1 = 7/3 - 3/3 = 4/3

So, the equation becomes:

27^(4/3) = 81

Now, we know that 27 = 3^3, so:

(3³⁾(4/3) = 81

Applying the power of a power rule:

3^(3*(4/3)) = 81

3^4 = 81

81 = 81

The equation holds true, so our solution x = -7/3 is correct. Verification is a vital step in any mathematical problem-solving process. It provides a check and balance, ensuring that the solution obtained is accurate. By substituting the value back into the original equation, we confirm that it satisfies the equation's conditions. This step not only gives us confidence in our answer but also reinforces our understanding of the problem and the solution process. It’s a best practice that saves time and prevents errors in the long run.

Conclusion

In this article, we've successfully solved the exponential equation 27^(-x-1) = 81. We started by understanding the core principle of expressing both sides of the equation with a common base. This allowed us to simplify the equation using the power of a power rule and then equate the exponents. Finally, we solved the resulting linear equation and verified our solution.

Solving exponential equations is a fundamental skill in mathematics, with applications across various fields. By mastering the techniques discussed here, you'll be well-equipped to tackle more complex problems. Remember, practice makes perfect, so keep solving and exploring different types of equations to strengthen your understanding.

This journey through solving exponential equations not only enhances your mathematical skills but also sharpens your problem-solving abilities in general. The systematic approach of identifying common bases, applying exponent rules, and verifying solutions is a valuable framework that can be applied to many other mathematical challenges. Keep practicing, and you'll find that these types of problems become more intuitive and straightforward. Happy solving!

For further exploration and a deeper understanding of exponential equations, consider visiting reputable resources like Khan Academy's Exponential Equations Section.