QuestionIdentify the solution set of $3 \ln 4=2 \ln x$ . \{6\} $\{-8,8\}$ $\{8\}$

Studdy Solution

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.

STEP 12

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

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms