Question

Let \( a = \frac{(\log 4) (\log 5 - \log 2)}{(\log 25)(\log 8 - \log 4)} \). Then the value of \( 5^a \) is:

• A. \( 7 \)
• B. \( 5 \)
• C. \( 8 \)
• D. \( \frac{5}{2} \)

Hint:

Simplify logarithmic expressions using logarithmic properties like \( \log a - \log b = \log \frac{a}{b} \) and \( \log (a^n) = n \log a \).

Answer 1

The Correct Option is C

Step 1: Simplifying the given expression: \[ a = \frac{(\log 4)(\log 5 - \log 2)}{(\log 25)(\log 8 - \log 4)}. \] Using properties of logarithms: \[ \log 4 = 2 \log 2, \quad \log 25 = 2 \log 5, \quad \log 8 = 3 \log 2. \] Substitute these into the expression: \[ a = \frac{(2 \log 2)(\log 5 - \log 2)}{(2 \log 5)(3 \log 2 - 2 \log 2)} = \frac{(2 \log 2)(\log 5 - \log 2)}{2 \log 5 (\log 2)}. \] Simplifying further: \[ a = \frac{\log 5 - \log 2}{\log 5} = 1 - \frac{\log 2}{\log 5}. \] Thus, the value of \( 5_a \) is \( 8 \).