Solving 10^x = 45: A Step-by-Step Guide

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Alright, guys, let's dive into solving this exponential equation! We're tasked with finding the value of x that satisfies the equation 10^x = 45, and we need to round our final answer to the nearest hundredth. This involves using logarithms, which might sound intimidating, but I'll break it down into simple, easy-to-follow steps. So, grab your calculators and let's get started!

Understanding Exponential Equations

Before we jump into the solution, let's quickly recap what exponential equations are all about. An exponential equation is an equation in which the variable appears in an exponent. In our case, we have 10 raised to the power of x, which equals 45. The key to solving these equations is to isolate the exponential term and then use logarithms to bring the exponent down. Logarithms are the inverse operation of exponentiation, meaning they "undo" the exponent. Specifically, if we have a^b = c, then loga(c) = b. This relationship is crucial for solving our problem.

Now, why do we use logarithms? Because they allow us to transform an exponential equation into a linear equation, which is much easier to solve. Think of it as a magic trick: logarithms help us rewrite the equation in a way that x is no longer stuck in the exponent but is instead a regular variable we can isolate. We can use either common logarithms (base 10) or natural logarithms (base e), but for this problem, using common logarithms will be the most straightforward approach, as our base is already 10.

To summarize, remember that exponential equations have variables in the exponent, and logarithms are the tool we use to bring those variables down and solve for them. Keep this in mind as we move forward, and you'll see how easy it becomes to tackle these types of problems. With a solid understanding of these basics, you'll be solving exponential equations like a pro in no time!

Step-by-Step Solution

Okay, let's get our hands dirty and solve for x in the equation 10^x = 45. Here’s a breakdown of each step:

  1. Apply the Common Logarithm: The first step is to apply the common logarithm (log base 10) to both sides of the equation. This gives us log(10^x) = log(45).

  2. Use the Power Rule of Logarithms: The power rule of logarithms states that logb(a^c) = c * logb(a). Applying this rule to the left side of our equation, we get x * log(10) = log(45).

  3. Simplify: Since log(10) is equal to 1 (because 10^1 = 10), our equation simplifies to x = log(45).

  4. Calculate the Logarithm: Now, we need to find the value of log(45). Using a calculator, we find that log(45) ≈ 1.6532.

  5. Round to the Nearest Hundredth: Finally, we round our answer to the nearest hundredth. So, x ≈ 1.65.

And that's it! We've successfully solved for x. The value of x that satisfies the equation 10^x = 45, rounded to the nearest hundredth, is approximately 1.65. Wasn't that easier than you thought?

Let's recap the key steps we took:

  • Applied the common logarithm to both sides.
  • Used the power rule of logarithms to bring down the exponent.
  • Simplified the equation.
  • Calculated the logarithm using a calculator.
  • Rounded the result to the nearest hundredth.

By following these steps, you can solve a wide variety of exponential equations. Just remember to use the appropriate logarithm (common or natural) based on the base of the exponential term. Practice makes perfect, so try solving a few more examples to solidify your understanding. You'll become more confident with each problem you solve!

Alternative Method: Using Natural Logarithms

Now, let's explore another way to solve the same problem using natural logarithms (ln). While using common logarithms is more straightforward when the base is 10, understanding how to use natural logarithms provides flexibility and a deeper understanding of logarithmic functions. So, let's dive in!

  1. Apply the Natural Logarithm: Instead of applying the common logarithm, we apply the natural logarithm (ln) to both sides of the equation: ln(10^x) = ln(45).

  2. Use the Power Rule of Logarithms: Just like before, we use the power rule of logarithms to bring down the exponent: x * ln(10) = ln(45).

  3. Isolate x: To isolate x, we divide both sides by ln(10): x = ln(45) / ln(10).

  4. Calculate the Logarithms: Using a calculator, we find that ln(45) ≈ 3.80666 and ln(10) ≈ 2.30259.

  5. Divide and Round: Now, we divide the two values: x ≈ 3.80666 / 2.30259 ≈ 1.6532. Rounding to the nearest hundredth, we get x ≈ 1.65.

As you can see, we arrived at the same answer using natural logarithms! The key difference is that we had to divide ln(45) by ln(10) to isolate x, whereas with common logarithms, log(10) simply equals 1. This method reinforces the understanding that logarithms of different bases can be used to solve exponential equations, and the choice often depends on the specific problem and personal preference. Keep in mind that both methods are equally valid, and it's always good to have multiple tools in your mathematical toolbox!

Common Mistakes to Avoid

When solving exponential equations, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Incorrectly Applying Logarithm Rules: One of the most common mistakes is misapplying the rules of logarithms. Make sure you understand and correctly apply the power rule, product rule, and quotient rule. For example, don't confuse log(a + b) with log(a) + log(b), as they are not the same.

  • Forgetting to Apply the Logarithm to Both Sides: Always remember to apply the logarithm to both sides of the equation. Applying it to only one side will change the equation and lead to an incorrect answer.

  • Calculator Errors: Be careful when using a calculator to evaluate logarithms. Ensure you're using the correct base (common or natural) and that you're entering the numbers correctly. A small typo can result in a significantly different answer.

  • Rounding Too Early: Avoid rounding intermediate values during your calculations. Rounding too early can introduce errors that accumulate and affect the accuracy of your final answer. Always wait until the very end to round.

  • Misunderstanding the Base of the Logarithm: Pay attention to the base of the logarithm. If the base is not explicitly written, it's usually assumed to be 10 (common logarithm). Make sure you're using the correct base for your calculations.

By being aware of these common mistakes, you can avoid them and increase your chances of solving exponential equations accurately. Always double-check your work and be meticulous in your calculations.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

  1. Solve for x: 5^x = 75 (Round to the nearest hundredth).
  2. Solve for x: 2^x = 32 (This one doesn't require rounding!).
  3. Solve for x: 12^x = 200 (Round to the nearest hundredth).

Try solving these problems using both common and natural logarithms to get comfortable with both methods. Remember to follow the steps we discussed earlier and avoid the common mistakes. The more you practice, the more confident you'll become in solving exponential equations. Good luck, and happy solving!