Math Snap

STEP 1

What is this asking?
We need to find the value of $x$ that makes the equation $3 \ln 4 = 2 \ln x$ true.
Watch out!
Remember the logarithm power rule: $n \ln a = \ln (a^n)$.
Don't forget to check your answer!

STEP 2

1. Rewrite the equation using the power rule.
2. Simplify both sides.
3. Isolate $x$.
4. Check the solution.

STEP 3

We have $3 \ln 4$.
Using the power rule of logarithms, which says $n \ln a = \ln (a^n)$, we can rewrite this as $\ln (4^3)$.
Why? Because it helps simplify the equation and brings the coefficient of the logarithm up as an exponent.
This makes it easier to compare the arguments of the logarithms.

STEP 4

We have $2 \ln x$.
Using the power rule, we can rewrite this as $\ln (x^2)$.
This helps us get closer to solving for $x$.

STEP 5

Now, our equation looks like this: $\ln (4^3) = \ln (x^2)$.
Much better!

STEP 6

We have $\ln (4^3)$.
Since $4^3 = 4 \cdot 4 \cdot 4 = 64$, we have $\ln 64$.

STEP 7

Our equation now looks like this: $\ln 64 = \ln (x^2)$.
Almost there!

STEP 8

Since $\ln 64 = \ln (x^2)$, we know that the arguments of the logarithms must be equal.
This means $64 = x^2$.

STEP 9

To solve $64 = x^2$ for $x$, we take the square root of both sides.
This gives us $x = \pm \sqrt{64}$, which simplifies to $x = \pm 8$.
So, $x$ can be either 8 or -8.

STEP 10

If $x = 8$, our original equation becomes $3 \ln 4 = 2 \ln 8$.
Using the power rule, this simplifies to $\ln (4^3) = \ln (8^2)$, or $\ln 64 = \ln 64$.
This is true!

STEP 11

If $x = -8$, our original equation becomes $3 \ln 4 = 2 \ln (-8)$.
Uh oh!
We can't take the logarithm of a negative number!
So, $-8$ is not a valid solution.

The solution set is $\{8\}$.