If `log_(x)a, a^(x//2), log_(b)x` are in G.P. then x is equal to

Options

A. `log_(a) (log_(b)a)`
B. `(log(log a) - log(log b))/(log a)`
C. `log_(b) (log_(a)b)`
D. none of these

Answer

To solve the problem, we need to determine the value of \( x \) given that \( \log_{x} a \), \( a^{(x/2)} \), and \( \log_{b} x \) are in geometric progression (G.P.).

### Step-by-Step Solution:

1. **Understanding the condition of G.P.**:
For three terms \( A, B, C \) to be in G.P., the square of the middle term must be equal to the product of the other two terms:
\[ B^2 = A \cdot C \]
In our case, let:
- \( A = \log_{x} a \)
- \( B = a^{(x/2)} \)
- \( C = \log_{b} x \)

Therefore, we have:
\[ (a^{(x/2)})^2 = \log_{x} a \cdot \log_{b} x \]

2. **Expanding the left side**:
The left side becomes:
\[ a^{x} = \log_{x} a \cdot \log_{b} x \]

3. **Using the change of base formula**:
We can express \( \log_{x} a \) using the change of base formula:
\[ \log_{x} a = \frac{\log a}{\log x} \]
and \( \log_{b} x \) as:
\[ \log_{b} x = \frac{\log x}{\log b} \]

Substituting these into our equation gives:
\[ a^{x} = \left(\frac{\log a}{\log x}\right) \cdot \left(\frac{\log x}{\log b}\right) \]

4. **Simplifying the equation**:
The equation simplifies to:
\[ a^{x} = \frac{\log a}{\log b} \]

5. **Taking logarithm on both sides**:
Taking logarithm (base 10 or natural log) on both sides:
\[ \log(a^{x}) = \log\left(\frac{\log a}{\log b}\right) \]
This simplifies to:
\[ x \log a = \log(\log a) - \log(\log b) \]

6. **Solving for \( x \)**:
Rearranging gives:
\[ x = \frac{\log(\log a) - \log(\log b)}{\log a} \]

Thus, the value of \( x \) is:
\[ x = \frac{\log(\log a) - \log(\log b)}{\log a} \]