Solving Exponential Equations A Step-by-Step Guide For 3^(2x-10) = 6^(4x-7)

Hey there, math enthusiasts! Today, we're diving into the exciting world of exponential equations. Specifically, we're going to tackle the equation 3^(2x-10) = 6^(4x-7). Exponential equations, at first glance, might seem a bit intimidating with those variables hanging out in the exponents. But fear not! With a systematic approach and a solid understanding of logarithmic properties, we can crack this equation and find the value of x. This guide is designed to walk you through each step, ensuring you grasp the underlying concepts and can confidently solve similar problems in the future. So, grab your calculators, and let's get started!

1. Understanding Exponential Equations

Before we jump into the solution, let's take a moment to understand what makes an equation exponential. In simple terms, an exponential equation is one where the variable appears in the exponent. Our equation, 3^(2x-10) = 6^(4x-7), perfectly fits this description. The key to solving these equations lies in our ability to manipulate them using logarithmic properties. Logarithms, in essence, are the inverse operation of exponentiation. Think of it this way: if exponentiation is like raising a number to a power, logarithms are like asking, "To what power must we raise this base to get this number?" Mastering logarithms is crucial for navigating the realm of exponential equations.

The Power of Logarithms

Logarithms provide us with a powerful tool to "bring down" exponents. The most important property we'll use here is the power rule of logarithms, which states that log_b(a^c) = c * log_b(a). This rule allows us to transform an exponent into a coefficient, making the variable x much easier to isolate and solve for. There are two main types of logarithms we often encounter: the common logarithm (log base 10) and the natural logarithm (log base e, denoted as ln). Both work equally well for solving exponential equations, so the choice often comes down to personal preference or calculator availability.

Why This Equation Matters

Now, you might be wondering, "Why should I care about solving this particular equation?" Well, exponential equations aren't just abstract mathematical concepts; they pop up in various real-world scenarios. From modeling population growth and radioactive decay to calculating compound interest and analyzing electrical circuits, exponential functions are fundamental tools in science, engineering, and finance. By mastering the techniques to solve equations like 3^(2x-10) = 6^(4x-7), you're not just honing your math skills; you're equipping yourself with a valuable problem-solving toolkit applicable across numerous disciplines. So, let's delve deeper and unlock the secrets to solving this equation!

2. Applying Logarithms to Both Sides

The first step in solving 3^(2x-10) = 6^(4x-7) is to apply a logarithm to both sides of the equation. This is a crucial step because it allows us to utilize the power rule of logarithms, which, as we discussed, is key to bringing those exponents down. The choice of logarithm base is arbitrary; you can use the common logarithm (log base 10), the natural logarithm (ln), or any other base. However, using either log base 10 or ln is generally preferred because most calculators have these functions readily available. For this example, let's use the natural logarithm (ln), but keep in mind that the process would be virtually identical if we used log base 10.

Why Natural Logarithm?

You might be curious why we're opting for the natural logarithm. Well, the natural logarithm has a special connection to the exponential function with base e (Euler's number, approximately 2.71828). This connection makes it particularly useful in calculus and other advanced mathematical contexts. However, for the purpose of solving this equation, the natural logarithm simply provides a convenient and widely supported option. The key takeaway here is that the underlying principle remains the same regardless of the base you choose.

Taking the Natural Logarithm

Applying the natural logarithm to both sides of our equation, 3^(2x-10) = 6^(4x-7), gives us:

ln(3^(2x-10)) = ln(6^(4x-7))

This might seem like a simple step, but it's a pivotal one. We've transformed the equation into a form where we can now leverage the power rule of logarithms. Notice how the exponents are still "up there," but we've set the stage for them to come down and become coefficients.

The Next Step: Unleashing the Power Rule

With the natural logarithm applied, we're now perfectly positioned to use the power rule. In the next section, we'll delve into how this rule transforms our equation and brings us closer to isolating the variable x. Remember, the goal is to manipulate the equation strategically, step by step, until we have x all by itself on one side. This process requires a solid understanding of logarithmic properties and a bit of algebraic finesse. So, let's continue our journey and unlock the next stage in solving this exponential equation!

3. Utilizing the Power Rule of Logarithms

Now, the moment we've been waiting for! This is where the power rule of logarithms truly shines. As a reminder, the power rule states that ln(a^c) = c * ln(a). We can apply this rule to both sides of our equation, ln(3^(2x-10)) = ln(6^(4x-7)), to bring the exponents down as coefficients. This is a game-changer because it transforms the exponential equation into a linear equation, which is much easier to solve.

Applying the Power Rule

Let's apply the power rule to the left side of the equation, ln(3^(2x-10)). According to the rule, we can move the exponent (2x-10) to the front as a coefficient, giving us:

(2x - 10) * ln(3)

Similarly, applying the power rule to the right side of the equation, ln(6^(4x-7)), gives us:

(4x - 7) * ln(6)

So, our equation now looks like this:

(2x - 10) * ln(3) = (4x - 7) * ln(6)

The Transformation: Exponential to Linear

Notice the dramatic shift! We've successfully transformed an exponential equation into a linear equation. The variable x is no longer trapped in the exponent; it's now part of a simple algebraic expression. This is a testament to the power of logarithms. By strategically applying the power rule, we've unlocked a pathway to solve for x.

The Road Ahead: Algebraic Manipulation

With the equation now in a linear form, our next step is to perform some algebraic manipulations to isolate x. This will involve expanding the expressions, collecting like terms, and ultimately solving for x. The key here is to be methodical and pay close attention to the order of operations. In the next section, we'll break down these algebraic steps and guide you through the process of isolating x. So, let's keep moving forward and bring this solution home!

4. Algebraic Manipulation and Isolation of x

With our equation transformed into (2x - 10) * ln(3) = (4x - 7) * ln(6), it's time to put our algebraic skills to work. The goal here is to isolate x, which means getting x all by itself on one side of the equation. This involves a series of steps: expanding the expressions, collecting terms with x on one side, and then solving for x. Let's break down each of these steps.

Expanding the Expressions

First, we need to expand both sides of the equation by distributing the logarithms. On the left side, we have (2x - 10) * ln(3), which expands to:

2x * ln(3) - 10 * ln(3)

On the right side, we have (4x - 7) * ln(6), which expands to:

4x * ln(6) - 7 * ln(6)

So, our equation now looks like this:

2x * ln(3) - 10 * ln(3) = 4x * ln(6) - 7 * ln(6)

Collecting Like Terms

The next step is to collect all the terms containing x on one side of the equation and all the constant terms on the other side. Let's move the x terms to the right side and the constant terms to the left side. To do this, we'll subtract 2x * ln(3) from both sides and add 7 * ln(6) to both sides. This gives us:

-10 * ln(3) + 7 * ln(6) = 4x * ln(6) - 2x * ln(3)

Now, let's factor out x from the right side:

-10 * ln(3) + 7 * ln(6) = x * (4 * ln(6) - 2 * ln(3))

Solving for x

We're almost there! To isolate x, we simply need to divide both sides of the equation by the expression in the parentheses:

x = (-10 * ln(3) + 7 * ln(6)) / (4 * ln(6) - 2 * ln(3))

This is the exact solution for x. It looks a bit messy, but it's a precise representation of the value of x that satisfies the original equation.

Approximating the Solution

For practical purposes, we often want to approximate the solution as a decimal. Using a calculator, we can evaluate the expression above to get an approximate value for x. Plugging in the values for ln(3) and ln(6), we find that:

x ≈ (-10 * 1.0986 + 7 * 1.7918) / (4 * 1.7918 - 2 * 1.0986)

x ≈ (1.5426) / (4.9700)

x ≈ 0.3104

So, the approximate solution for x is about 0.3104. This means that if we plug this value back into the original equation, 3^(2x-10) = 6^(4x-7), both sides should be approximately equal.

The Importance of Precision

While the approximate solution is useful for understanding the magnitude of x, it's important to remember that the exact solution is the most accurate representation. In some contexts, even small differences in the value of x can have significant impacts. Therefore, it's always a good practice to keep the exact solution in its logarithmic form unless a decimal approximation is specifically required.

5. Verification and Conclusion

We've journeyed through the process of solving the exponential equation 3^(2x-10) = 6^(4x-7), and we've arrived at a solution. But before we declare victory, it's always a good idea to verify our solution. This helps us ensure that we haven't made any mistakes along the way and that our answer truly satisfies the original equation. Let's walk through the verification process.

Plugging the Solution Back In

The most straightforward way to verify our solution is to plug it back into the original equation. We have the exact solution:

x = (-10 * ln(3) + 7 * ln(6)) / (4 * ln(6) - 2 * ln(3))

And the approximate solution:

x ≈ 0.3104

Let's use the approximate solution for verification purposes, as it's easier to work with in this context. Plugging x ≈ 0.3104 into the left side of the equation, 3^(2x-10), gives us:

3^(2 * 0.3104 - 10) ≈ 3^(-9.3792)

Using a calculator, we find that 3^(-9.3792) ≈ 0.00002009

Now, let's plug x ≈ 0.3104 into the right side of the equation, 6^(4x-7), which gives us:

6^(4 * 0.3104 - 7) ≈ 6^(-5.7584)

Using a calculator, we find that 6^(-5.7584) ≈ 0.00002009

Comparing the Results

Notice that both sides of the equation yield approximately the same value: 0.00002009. This strongly suggests that our solution, x ≈ 0.3104, is correct. Keep in mind that due to rounding, there might be slight discrepancies in the values, but they should be very close.

Why Verification Matters

Verification is a crucial step in any mathematical problem-solving process. It acts as a safety net, catching potential errors in our calculations or algebraic manipulations. By plugging our solution back into the original equation, we gain confidence that our answer is indeed correct. This is particularly important in more complex problems where the chances of making a mistake are higher.

Key Takeaways

Throughout this guide, we've explored the process of solving the exponential equation 3^(2x-10) = 6^(4x-7). We've learned the importance of logarithms, the power rule of logarithms, and the algebraic techniques required to isolate the variable x. We've also emphasized the significance of verifying our solution to ensure accuracy. By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of exponential equations.

Final Thoughts

Solving exponential equations might seem challenging at first, but with a systematic approach and a solid understanding of the underlying principles, it becomes a manageable task. Remember to break down the problem into smaller steps, utilize the power of logarithms, and always verify your solution. With practice, you'll become more confident and proficient in solving these types of equations. So, keep exploring, keep learning, and keep pushing your mathematical boundaries! You've got this!