Solving 7^(3x-2) = 342^(2x+5): A Step-by-Step Guide

Hey guys, ever stumbled upon an equation that looks like it's speaking a different language? Well, exponential equations can feel just like that! But don't worry, we're about to break down one of these bad boys step-by-step. Let's tackle the equation 7^(3x-2) = 342^(2x+5). It looks intimidating, but trust me, we'll conquer it together.

Understanding the Challenge: Exponential Equations

So, what exactly makes this an exponential equation? It's all about the variable, x, chilling out in the exponent. Our mission is to isolate x and figure out its value. The main hurdle here is that x is part of a power, and we need to bring it down to ground level to solve for it. To solve this equation, we'll need to use logarithms, which are the inverse operations of exponentiation. Logarithms allow us to "undo" the exponent and bring the variable down to a level where we can work with it more easily. The equation we're dealing with, 7^(3x-2) = 342^(2x+5), presents a unique challenge because the bases (7 and 342) are different. This means we can't simply equate the exponents. Instead, we need to apply logarithms to both sides of the equation. The choice of base for the logarithm is crucial; while any base will technically work, using a common base like 10 or the natural logarithm (base e) makes the calculations more straightforward and easier to interpret. By applying the logarithm, we can use the power rule of logarithms, which states that log_b(a^c) = c * log_b(a). This rule is essential because it allows us to move the exponents (3x-2) and (2x+5) from the power to the coefficient, bringing x out of the exponent and into a position where we can solve for it. This transformation is the key step in solving exponential equations.

Step 1: Applying Logarithms to Both Sides

The golden rule of equation solving? Whatever you do to one side, you gotta do to the other! In this case, we're going to apply the logarithm to both sides of the equation. We'll use the common logarithm (base 10) for simplicity, but you could totally use the natural logarithm (ln) if you prefer. Remember, the goal here is to bring those exponents down. By applying the logarithm to both sides, we maintain the balance of the equation while setting ourselves up to use the power rule of logarithms. This step is crucial because it allows us to manipulate the equation in a way that isolates x. So, let's get to it and apply the logarithm to both sides, setting the stage for the next steps in solving this exponential equation. Don't worry, we're making progress, guys! The logarithmic function, denoted as log_b(x), is the inverse of the exponential function. It answers the question: "To what power must we raise the base b to get x?" In our case, applying the logarithm helps us unravel the exponent and bring the variable x into the spotlight. So, let's write it out: log(7^(3x-2)) = log(342^(2x+5)). See how we've applied the log to both sides? Now, we're ready for the next powerful move.

Step 2: Unleashing the Power Rule of Logarithms

Now, for the magic trick! The power rule of logarithms is our best friend here. It states that log_b(a^c) = c * log_b(a). In plain English, it means we can take that exponent and move it to the front as a multiplier. This is exactly what we need to do to get x out of the exponent's clutches. This is where the equation really starts to transform into something we can handle. By applying the power rule, we're essentially simplifying the equation and bringing the variable x into a position where we can isolate it. So, get ready to see some algebraic action as we unleash the power rule and move closer to the solution. The power rule is a fundamental property of logarithms that makes them incredibly useful for solving exponential equations. It allows us to rewrite the equation in a linear form, which is much easier to manipulate algebraically. By understanding and applying this rule, we can transform complex exponential equations into simpler, more manageable expressions. So, let's take a deep breath and apply the power rule to both sides of our equation. Remember, we're not just moving numbers around; we're unlocking the secrets hidden within the equation. Let's rewrite our equation using the power rule: (3x - 2) * log(7) = (2x + 5) * log(342). Bam! Look at that. The exponents are now coefficients, and x is finally on the same level as the other terms. We're making serious progress, guys!

Step 3: Expanding and Rearranging the Equation

Alright, time to roll up our sleeves and get algebraic! We need to expand both sides of the equation by distributing the logarithms. This means multiplying log(7) by both 3x and -2, and doing the same with log(342) and 2x + 5. Think of it like unwrapping a present – we're revealing the individual terms that make up the equation. Expanding the equation is a crucial step because it allows us to separate the terms containing x from the constant terms. This separation is necessary to isolate x and ultimately solve for its value. So, let's carefully distribute the logarithms and see what new form our equation takes. Remember, each term plays a vital role in the equation, and by expanding, we're making sure we account for every piece of the puzzle. So, let's get those terms expanded and prepare for the next stage of rearranging the equation. This process might seem a bit tedious, but it's a necessary step in solving for x. By carefully distributing and expanding, we're setting ourselves up for success. Let's do it: 3x * log(7) - 2 * log(7) = 2x * log(342) + 5 * log(342). Now, it looks a bit messier, but we're actually closer to our goal. Our next move is to gather all the x terms on one side and all the constant terms on the other. This is a classic algebraic maneuver that helps us isolate the variable we're trying to solve for. Let's move all the terms with x to the left side and the constant terms to the right side. Remember, when moving a term from one side to the other, we change its sign. This is a fundamental principle of algebra that we need to keep in mind to avoid errors. So, let's carefully rearrange the terms and see how our equation shapes up. We're getting closer to the final solution, guys! So, let's rearrange and see the magic happen: 3x * log(7) - 2x * log(342) = 5 * log(342) + 2 * log(7). Phew! We've successfully gathered like terms. Now, we're ready to factor out x on the left side.

Step 4: Isolating x – The Grand Finale!

We're in the home stretch now! The left side has two terms with x in them, so let's factor that x out. This is like putting x in a spotlight, making it the star of the show. Factoring out x is a key step in isolating the variable. It allows us to combine the terms containing x into a single expression, making it easier to solve for x. So, let's factor out x and see how much simpler our equation becomes. Remember, the goal here is to get x all by itself on one side of the equation. By factoring, we're taking a big step towards achieving that goal. So, let's factor out x and prepare for the final division. We're so close to the solution, guys! By factoring out x, we're essentially undoing the distribution we did earlier. It's like rewinding the process to get back to the core of the equation. So, let's do it: x * (3 * log(7) - 2 * log(342)) = 5 * log(342) + 2 * log(7). Almost there! Now, we have x multiplied by a whole bunch of logarithms. To finally isolate x, we'll divide both sides of the equation by that bunch. This will leave x all alone on the left side, and we'll have our solution on the right side. Dividing both sides by the same quantity is another fundamental principle of algebra that ensures we maintain the balance of the equation. So, let's take a deep breath and perform the final division. We're about to reveal the value of x, guys! So, let's divide and conquer: x = (5 * log(342) + 2 * log(7)) / (3 * log(7) - 2 * log(342)).

Step 5: Calculating the Solution (with a calculator!)

Okay, let's be real – we're not going to calculate those logarithms by hand! Grab your calculator (or your favorite online calculator) and plug in the values. Make sure you're using the same base logarithm you used throughout the problem (we used base 10). This is where the numerical magic happens. After all the algebraic manipulation, we finally get to see the numerical value of x. So, let's carefully plug in the values into our calculator and see what we get. Remember to follow the order of operations (PEMDAS/BODMAS) to ensure accurate calculations. So, let's get those numbers crunched and reveal the solution. We're almost there, guys! Using a calculator, we find that log(342) is approximately 2.534 and log(7) is approximately 0.845. Substituting these values into our equation, we get: x ≈ (5 * 2.534 + 2 * 0.845) / (3 * 0.845 - 2 * 2.534). Now, let's do the arithmetic: x ≈ (12.67 + 1.69) / (2.535 - 5.068) x ≈ 14.36 / -2.533 x ≈ -5.67. So, there you have it! The solution to the equation 7^(3x-2) = 342^(2x+5) is approximately x = -5.67. We conquered that exponential beast!

Conclusion: Exponential Equations, No Problem!

See? Exponential equations aren't so scary after all! By using logarithms and the power rule, we can solve even the trickiest ones. The key is to break it down step-by-step and stay organized. Remember, practice makes perfect, so keep those equations coming! Guys, you've got this! Remember, the key to solving exponential equations is to understand the properties of logarithms and apply them strategically. Each step, from applying the logarithm to both sides to isolating x, is a deliberate move towards the final solution. By breaking down the problem into smaller, manageable steps, we can tackle even the most daunting equations. So, don't be intimidated by those exponents; embrace the challenge and remember that with the right tools and techniques, you can conquer any exponential equation that comes your way. Keep practicing, keep exploring, and you'll become a master of exponential equations in no time! And hey, if you ever get stuck, just remember this journey we took together, step-by-step, and you'll find your way to the solution. You've got this, guys!