Solving Exponential Equations With Common Logarithms

Hey guys! Today, we're diving into how to solve exponential equations using common logarithms. Exponential equations might seem intimidating at first, but with the right tools and a bit of practice, you'll be solving them like a pro. We'll specifically tackle an equation where we need to use the common logarithm to isolate the variable. So, grab your calculators, and let's get started!

Understanding Exponential Equations and Logarithms

Before we jump into solving, let's make sure we're all on the same page. An exponential equation is an equation where the variable appears in the exponent. For example, $5^{5x+3} = 39$ is an exponential equation. Our goal is to isolate x, but it's stuck up there in the exponent. That's where logarithms come to the rescue!

A logarithm is the inverse operation of exponentiation. In simpler terms, it helps us find the exponent needed to raise a base to a certain power. The expression $\log_b(a) = c$ means that $b^c = a$. Here, b is the base, a is the argument, and c is the exponent. When we talk about the common logarithm, we're referring to a logarithm with a base of 10. It's written as $\log_{10}(x)$ or simply $\log(x)$. Most calculators have a built-in function for calculating common logarithms, making them super handy for solving these types of equations. So, remember, when you see "log" without a base specified, it's almost always referring to the common logarithm (base 10). This is crucial for using your calculator correctly, especially when approximating solutions to a specific number of decimal places. Understanding this relationship between exponents and logarithms is fundamental to solving exponential equations effectively.

Rewriting the Exponential Equation Using the Common Logarithm

Okay, let's get back to our equation: $5^{5x+3} = 39$. The key to solving this using logarithms is to take the common logarithm of both sides. This allows us to bring the exponent down using a logarithm property. Here’s how it works:

1. Take the common logarithm of both sides: $\log(5^{5x+3}) = \log(39)$
2. Use the power rule of logarithms: The power rule states that $\log_b(a^c) = c \cdot \log_b(a)$. Applying this rule to our equation, we get: $(5x + 3) \cdot \log(5) = \log(39)$

Now, notice that the exponent $(5x + 3)$ is no longer an exponent! It's now a coefficient, which means we can work with it using regular algebraic techniques. This is the magic of logarithms – they transform exponential problems into linear problems. By applying the common logarithm, we've effectively "unlocked" the variable x from its exponential prison. Keep in mind that $\log(5)$ and $\log(39)$ are just numbers, which your calculator can easily compute. This step is crucial because it sets us up to isolate x and find its approximate value. So, remember to take the logarithm of both sides and use the power rule to bring down the exponent; it’s the cornerstone of solving exponential equations using logarithms. Without this step, isolating the variable would be virtually impossible!

Isolating the Variable

Now that we've rewritten the equation as $(5x + 3) \cdot \log(5) = \log(39)$, our next step is to isolate x. This involves a bit of algebraic manipulation, but don't worry, we'll take it step by step:

1. Divide both sides by $\log(5)$: To get $(5x + 3)$ by itself, we divide both sides of the equation by $\log(5)$: $5x + 3 = \frac{\log(39)}{\log(5)}$
2. Subtract 3 from both sides: Next, we subtract 3 from both sides to isolate the term with x: $5x = \frac{\log(39)}{\log(5)} - 3$
3. Divide both sides by 5: Finally, we divide both sides by 5 to solve for x: $x = \frac{\frac{\log(39)}{\log(5)} - 3}{5}$

So, we've successfully isolated x! The expression $x = \frac{\frac{\log(39)}{\log(5)} - 3}{5}$ gives us the exact solution to the equation. However, it's not very practical in this form. We need to use a calculator to approximate the value of x to three decimal places. Remember, each of these steps is crucial for isolating the variable. Dividing by $\log(5)$, subtracting 3, and dividing by 5 are all necessary to get x by itself. If you skip or misapply any of these steps, you'll end up with the wrong answer. So, take your time, follow the order of operations, and double-check your work to ensure you're isolating the variable correctly!

Approximating the Solution with a Calculator

Now that we have $x = \frac{\frac{\log(39)}{\log(5)} - 3}{5}$, it's time to use a calculator to find the approximate value of x to three decimal places. Here’s how you can do it:

1. Calculate $\log(39)$ and $\log(5)$: Using your calculator, find the common logarithms of 39 and 5: $\log(39) \approx 1.59106$ $\log(5) \approx 0.69897$
2. Divide $\log(39)$ by $\log(5)$: $\frac{\log(39)}{\log(5)} \approx \frac{1.59106}{0.69897} \approx 2.27627$
3. Subtract 3: $2.27627 - 3 = -0.72373$
4. Divide by 5: $\frac{-0.72373}{5} = -0.144746$
5. Round to three decimal places: Rounding -0.144746 to three decimal places gives us: $x \approx -0.145$

Therefore, the approximate solution to the equation $5^{5x+3} = 39$, rounded to three decimal places, is x ≈ -0.145. This final step is where your calculator skills really shine. Make sure you enter the values correctly and follow the order of operations to get an accurate approximation. Also, double-check your rounding to ensure you're providing the answer to the specified number of decimal places. Accuracy is key here, so take your time and use your calculator wisely! This approximation is what we need as the solution.

Checking the Solution

To ensure our solution is correct, we can plug the approximate value of x back into the original equation and see if it holds true. Let's substitute x ≈ -0.145 into $5^{5x+3}$:

$5^{5(-0.145)+3} = 5^{-0.725+3} = 5^{2.275} \approx 38.93$

Since 38.93 is very close to 39, we can be confident that our solution x ≈ -0.145 is accurate to three decimal places. This check is a great way to catch any errors you might have made along the way. If the result is significantly different from 39, it's a sign that you need to go back and review your calculations. So, always remember to check your solution to ensure it's accurate and reliable!

Conclusion

So, there you have it! We've successfully solved the exponential equation $5^{5x+3} = 39$ using the common logarithm and approximated the solution to three decimal places. Remember, the key steps are taking the common logarithm of both sides, using the power rule to bring down the exponent, isolating the variable, and using a calculator to approximate the solution. With practice, you'll become more comfortable with these steps and be able to solve exponential equations with ease. Keep practicing, and you'll master these skills in no time! Keep solving, guys!