Question

Find the value of \(\log_{3{^2}}5^4\times\log_{5{^2}}3^4\)

• A. 5
• B. 3
• C. 4
• D. 2

Answer 1

The Correct Option is C

To solve the problem, we need to find the value of \(\log_{3^2}5^4 \times \log_{5^2}3^4\). Let's denote the expression as \(A\) and analyze it step by step.

\(A = \log_{3^2}5^4 \times \log_{5^2}3^4\)

We can use the change of base formula:

\(\log_{a^b}c = \frac{\log c}{b\log a}\)

Therefore:

\(\log_{3^2}5^4 = \frac{\log 5^4}{2\log 3} = \frac{4\log 5}{2\log 3} = 2\frac{\log 5}{\log 3}\)

\(\log_{5^2}3^4 = \frac{\log 3^4}{2\log 5} = \frac{4\log 3}{2\log 5} = 2\frac{\log 3}{\log 5}\)

Now substituting these back into \(A\):

\(A = \left(2\frac{\log 5}{\log 3}\right) \times \left(2\frac{\log 3}{\log 5}\right)\)

\(A = 4\frac{\log 5}{\log 3} \cdot \frac{\log 3}{\log 5}\)

As the terms \(\frac{\log 5}{\log 3}\) and \(\frac{\log 3}{\log 5}\) multiply to 1, we have:

\(A = 4 \times 1 = 4\)

Thus, the value is 4.