The value of 'a' for which `(log_(a)7)/(log_(6)7) log_(pi) 36` holds good, is

Options

A. `1//pi`
B. `pi^(2)`
C. `sqrt(pi)`
D. 2

Answer

To solve the problem, we need to find the value of 'a' for which the equation

\[ \frac{\log_a 7}{\log_6 7} = \log_\pi 36 \]

holds true.

### Step 1: Rewrite the logarithmic expressions using the change of base formula.

Using the change of base formula, we can rewrite the logarithms:

\[ \log_a 7 = \frac{\log 7}{\log a} \quad \text{and} \quad \log_6 7 = \frac{\log 7}{\log 6} \]

Substituting these into the equation gives us:

\[ \frac{\frac{\log 7}{\log a}}{\frac{\log 7}{\log 6}} = \log_\pi 36 \]

### Step 2: Simplify the left-hand side.

The left-hand side simplifies as follows:

\[ \frac{\log 7}{\log a} \cdot \frac{\log 6}{\log 7} = \frac{\log 6}{\log a} \]

So, we have:

\[ \frac{\log 6}{\log a} = \log_\pi 36 \]

### Step 3: Rewrite the right-hand side using the change of base formula.

Using the change of base formula again, we can rewrite \(\log_\pi 36\):

\[ \log_\pi 36 = \frac{\log 36}{\log \pi} \]

### Step 4: Set the two sides equal to each other.

Now we can set the two sides equal:

\[ \frac{\log 6}{\log a} = \frac{\log 36}{\log \pi} \]

### Step 5: Cross-multiply to eliminate the fractions.

Cross-multiplying gives us:

\[ \log 6 \cdot \log \pi = \log 36 \cdot \log a \]

### Step 6: Solve for \(\log a\).

Rearranging the equation to solve for \(\log a\):

\[ \log a = \frac{\log 6 \cdot \log \pi}{\log 36} \]

### Step 7: Substitute \(\log 36\) with \(\log(6^2)\).

Since \(36 = 6^2\), we can rewrite \(\log 36\):

\[ \log 36 = 2 \log 6 \]

Substituting this back into our equation gives:

\[ \log a = \frac{\log 6 \cdot \log \pi}{2 \log 6} \]

### Step 8: Simplify the expression.

The \(\log 6\) terms cancel out (assuming \(\log 6 \neq 0\)):

\[ \log a = \frac{\log \pi}{2} \]

### Step 9: Exponentiate to solve for \(a\).

Exponentiating both sides gives:

\[ a = 10^{\frac{\log \pi}{2}} = \pi^{\frac{1}{2}} = \sqrt{\pi} \]

### Conclusion

Thus, the value of \(a\) is:

\[ \boxed{\sqrt{\pi}} \]