Solving Exponential Equations: Find X Where 3^(2x) = 9^(3x-4)

Hey guys! Let's dive into solving an interesting exponential equation. Our mission, should we choose to accept it, is to find the value of x that satisfies the equation $3^{2x} = 9^{3x-4}$. Don't worry; it's not as intimidating as it looks! We'll break it down step by step, so it's super easy to follow along. Exponential equations pop up all the time in math, science, and even finance, so mastering these skills is a total win. The key to cracking this problem lies in recognizing that we can express both sides of the equation with the same base. This allows us to equate the exponents and solve for x. Ready to jump in and get started? Let’s do it!

Step-by-Step Solution

1. Express Both Sides with the Same Base

The first thing we need to do is rewrite the equation so that both sides have the same base. Notice that 9 is a power of 3; specifically, $9 = 3^2$. So, we can rewrite the right side of the equation as follows:

$9^{3x-4} = (3^2)^{3x-4}$

Using the power of a power rule, which states that $(a^m)^n = a^{mn}$, we simplify the right side further:

$(3^2)^{3x-4} = 3^{2(3x-4)}$

So now, our original equation $3^{2x} = 9^{3x-4}$ becomes:

$3^{2x} = 3^{2(3x-4)}$

2. Equate the Exponents

Now that both sides of the equation have the same base (which is 3), we can equate the exponents. This means we set the exponents equal to each other:

$2x = 2(3x-4)$

3. Solve for x

Next up, we solve the resulting linear equation for x. First, distribute the 2 on the right side:

$2x = 6x - 8$

Now, let's get all the x terms on one side of the equation. Subtract $6x$ from both sides:

$2x - 6x = -8$

$-4x = -8$

Finally, divide both sides by -4 to solve for x:

$x = \frac{-8}{-4}$

$x = 2$

So, the value of x that satisfies the equation $3^{2x} = 9^{3x-4}$ is x = 2. Awesome job! You did it.

Verification

To make sure our answer is correct, let's plug x = 2 back into the original equation and see if both sides are equal:

Original equation: $3^{2x} = 9^{3x-4}$

Substitute x = 2:

$3^{2(2)} = 9^{3(2)-4}$

$3^4 = 9^{6-4}$

$3^4 = 9^2$

$81 = 81$

Since both sides of the equation are equal, our solution x = 2 is indeed correct! Fantastic! Always good to double-check.

Alternative Method: Using Logarithms (Just for Fun!)

While equating exponents is the most straightforward method for this particular problem, you can also use logarithms to solve for x. This method is especially useful when you can't easily express both sides with the same base. Let's walk through it quickly.

Starting with the original equation:

$3^{2x} = 9^{3x-4}$

Take the logarithm of both sides. You can use any base for the logarithm, but the natural logarithm (ln) or the common logarithm (log base 10) are often convenient. Let's use the natural logarithm (ln):

$ln(3^{2x}) = ln(9^{3x-4})$

Using the power rule for logarithms, which states that $ln(a^b) = b \cdot ln(a)$, we get:

$2x \cdot ln(3) = (3x-4) \cdot ln(9)$

Now, remember that $9 = 3^2$, so $ln(9) = ln(3^2) = 2 \cdot ln(3)$. Substitute this into the equation:

$2x \cdot ln(3) = (3x-4) \cdot 2 \cdot ln(3)$

Divide both sides by $ln(3)$ (since $ln(3) \neq 0$):

$2x = (3x-4) \cdot 2$

$2x = 6x - 8$

This is the same linear equation we obtained before, and solving for x gives us x = 2. See? Logarithms can be another tool in your arsenal!

Common Mistakes to Avoid

When solving exponential equations, there are a few common mistakes to watch out for:

1. Forgetting to Distribute: Make sure you distribute correctly when dealing with expressions like $a(b+c)$. For example, in our problem, it's crucial to correctly distribute the 2 in $2(3x-4)$.

2. Incorrectly Applying Exponent Rules: Be careful when applying exponent rules like $(a^m)^n = a^{mn}$. A wrong application can lead to an incorrect equation.

3. Not Checking Your Answer: Always verify your solution by plugging it back into the original equation. This helps catch any errors you might have made along the way.

4. Assuming You Can Always Equate Exponents: You can only equate exponents when the bases are the same. If the bases are different, you'll need to use logarithms or find a way to express both sides with the same base.

Practice Problems

Want to sharpen your skills? Here are a few practice problems you can try:

1. Solve for x: $2^{3x} = 4^{x+1}$
2. Solve for x: $5^{x-2} = 25^{2x-3}$
3. Solve for x: $4^{2x+1} = 16^{x-1}$

These problems are similar to the one we just solved, so you can use the same techniques to find the solutions. Good luck, and have fun!

Conclusion

Alright, you've successfully learned how to solve for x in the exponential equation $3^{2x} = 9^{3x-4}$. Remember, the key is to express both sides of the equation with the same base, equate the exponents, and then solve the resulting linear equation. Always verify your answer to make sure it's correct. Keep practicing, and you'll become a pro at solving exponential equations in no time! You've got this, and I am so proud of your effort. Solving exponential equations might seem hard, but breaking them down makes things so much easier. Keep practicing, and you'll be solving even trickier problems in no time! Keep up the fantastic work, and I can’t wait to see what you conquer next!