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 find the value of \( a \) for which

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

holds true, we can follow these steps:

### Step 1: Rewrite the logarithmic expressions
Using the change of base formula, we can express the logarithms in terms of natural logarithms (or any common base):

\[ \frac{\log_a 7}{\log_6 7} = \frac{\frac{\log 7}{\log a}}{\frac{\log 7}{\log 6}} = \frac{\log 7}{\log a} \cdot \frac{\log 6}{\log 7} = \frac{\log 6}{\log a} \]

### Step 2: Set the equation equal to \(\log_\pi 36\)
Now we can rewrite the original equation:

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

Using the change of base formula on \(\log_\pi 36\):

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

Thus, we can set up the equation:

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

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:

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

### Step 4: Express \(\log 36\) in terms of \(\log 6\)
We know that \( 36 = 6^2 \), so we can express \(\log 36\) as:

\[ \log 36 = \log(6^2) = 2 \log 6 \]

Substituting this back into the equation gives:

\[ \log 6 \cdot \log \pi = 2 \log 6 \cdot \log a \]

### Step 5: Cancel \(\log 6\) (assuming \(\log 6 \neq 0\))
Dividing both sides by \(\log 6\) (since \(\log 6\) is positive and not zero):

\[ \log \pi = 2 \log a \]

### Step 6: Solve for \(\log a\)
Now we can isolate \(\log a\):

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

### Step 7: Exponentiate to solve for \(a\)
Taking the antilogarithm of both sides gives:

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

### Final Answer
Thus, the value of \( a \) is:

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