Find the value of `log_(3^(2))5^(4)xxlog_(5^(2))3^(4)` .

Options

A. A)5
B. B)3
C. C)4
D. D)2

Answer

To solve the expression \( \log_{3^2}(5^4) \times \log_{5^2}(3^4) \), we can follow these steps:

### Step 1: Rewrite the logarithms using the change of base formula
Using the property \( \log_a(b) = \frac{1}{\log_b(a)} \), we can rewrite the logarithms:
\[ \log_{3^2}(5^4) = \frac{\log_{5^2}(3^4)}{1} \]
Thus, we have:
\[ \log_{3^2}(5^4) = \frac{1}{\log_{5^2}(3^4)} \]

### Step 2: Substitute back into the expression
Now substituting this back into the original expression:
\[ \log_{3^2}(5^4) \times \log_{5^2}(3^4) = \frac{1}{\log_{5^2}(3^4)} \times \log_{5^2}(3^4) \]

### Step 3: Simplify the expression
The expression simplifies to:
\[ \frac{\log_{5^2}(3^4)}{\log_{5^2}(3^4)} = 1 \]

### Step 4: Final multiplication
Now, we multiply by the factor we initially separated out:
\[ 1 \times 1 = 1 \]

### Conclusion
Thus, the value of \( \log_{3^2}(5^4) \times \log_{5^2}(3^4) \) is \( 1 \).