The value of `a^(("log"_(b)("log"_(b)x))/("log"_(b) a))`, is

Options

A. `"log"_(a)x`
B. `"log"_(b)x`
C. `"log"_(x)a`
D. `"log"_(x)b`

Answer

To solve the expression \( a^{\frac{\log_b(\log_b x)}{\log_b a}} \), we will use properties of logarithms.

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ a^{\frac{\log_b(\log_b x)}{\log_b a}} \]

2. **Use the Change of Base Formula**:
Recall the change of base formula for logarithms:
\[ \frac{\log_b A}{\log_b B} = \log_B A \]
Applying this to our expression, we can rewrite it as:
\[ a^{\log_a(\log_b x)} \]

3. **Apply the Power of a Logarithm**:
We use the property that \( a^{\log_a x} = x \):
\[ a^{\log_a(\log_b x)} = \log_b x \]

4. **Final Result**:
Therefore, the value of the original expression simplifies to:
\[ \log_b x \]

### Conclusion:
The value of \( a^{\frac{\log_b(\log_b x)}{\log_b a}} \) is \( \log_b x \).

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