Sure, let's solve the equation step by step:
The given equation is:
[tex]\[
3 \log_4 x = \log_4 32 + \log_4 2
\][/tex]
### Step 1: Use the properties of logarithms
First, recall the property of logarithms that states:
[tex]\[
\log_b a + \log_b c = \log_b (a \cdot c)
\][/tex]
So, we can combine the logarithms on the right-hand side:
[tex]\[
\log_4 32 + \log_4 2 = \log_4 (32 \cdot 2)
\][/tex]
### Step 2: Simplify the right-hand side
Calculate [tex]\(32 \cdot 2\)[/tex]:
[tex]\[
32 \cdot 2 = 64
\][/tex]
Thus, the equation simplifies to:
[tex]\[
3 \log_4 x = \log_4 64
\][/tex]
### Step 3: Solve for [tex]\(\log_4 x\)[/tex]
Since the equation states that [tex]\(3 \log_4 x\)[/tex] equals [tex]\(\log_4 64\)[/tex], we can set up the following equation:
[tex]\[
3 \log_4 x = \log_4 64
\][/tex]
To isolate [tex]\(\log_4 x\)[/tex], divide both sides by 3:
[tex]\[
\log_4 x = \frac{\log_4 64}{3}
\][/tex]
### Step 4: Evaluate [tex]\(\log_4 64\)[/tex]
We know that [tex]\(64\)[/tex] is equal to [tex]\(4^3\)[/tex]:
[tex]\[
64 = 4^3
\][/tex]
Therefore:
[tex]\[
\log_4 64 = \log_4 (4^3) = 3
\][/tex]
### Step 5: Substitute back into the equation
Now we substitute [tex]\(\log_4 64\)[/tex] with [tex]\(3\)[/tex]:
[tex]\[
\log_4 x = \frac{3}{3} = 1
\][/tex]
### Step 6: Solve for [tex]\(x\)[/tex]
Finally, recall that [tex]\(\log_4 x = 1\)[/tex] implies:
[tex]\[
4^1 = x \implies x = 4
\][/tex]
### Conclusion
The solution to the equation [tex]\(3 \log_4 x = \log_4 32 + \log_4 2\)[/tex] is:
[tex]\[
x = 4
\][/tex]
Thus, the correct answer is [tex]\(x = 4\)[/tex].
