If `"log"_(3) a xx "log"_(a) x = 4`, then x is equal to

Options

A. 64
B. 81
C. `a^(2)`
D. 12

Explanation

To solve the equation \( \log_{3} a \cdot \log_{a} x = 4 \), we will follow these steps:

### Step 1: Rewrite the logarithmic expressions
We start with the equation:
\[ \log_{3} a \cdot \log_{a} x = 4 \]
Using the change of base formula, we can express \( \log_{3} a \) and \( \log_{a} x \) in terms of natural logarithms (or any common logarithm):
\[ \log_{3} a = \frac{\log a}{\log 3} \quad \text{and} \quad \log_{a} x = \frac{\log x}{\log a} \]
Substituting these into the equation gives:
\[ \frac{\log a}{\log 3} \cdot \frac{\log x}{\log a} = 4 \]

### Step 2: Simplify the equation
In the equation above, \( \log a \) in the numerator and denominator cancels out:
\[ \frac{\log x}{\log 3} = 4 \]

### Step 3: Cross-multiply to isolate \( \log x \)
Now, we can cross-multiply to isolate \( \log x \):
\[ \log x = 4 \cdot \log 3 \]

### Step 4: Rewrite the logarithmic equation in exponential form
Using the property of logarithms that states \( \log_b a = c \) implies \( a = b^c \), we rewrite the equation:
\[ x = 3^4 \]

### Step 5: Calculate the value of \( x \)
Now, we calculate \( 3^4 \):
\[ x = 81 \]

Thus, the value of \( x \) is \( 81 \).

### Final Answer
\[ \boxed{81} \]