Question

if \\(\log_{6}(25) = 2ab^{-1}\\), which of these can represent \\(a\\) and \\(b\\)?

• \\(a = \ln(6)\\) and \\(b = \ln(5)\\)
• \\(a = \ln(5)\\) and \\(b = \ln(6)\\)
• \\(a = \ln(25)\\) and \\(b = \ln(6)\\)
• \\(a = \ln(5)\\) and \\(b = \frac{1}{\ln 6}\\)

Explanation:

Simplify the given logarithmic expression

$$ \log_{6}(25) = \log_{6}(5^2) = 2\log_{6}(5) $$

Apply the change of base formula

$$ 2\log_{6}(5) = 2 \cdot \frac{\ln(5)}{\ln(6)} = 2\ln(5)(\ln(6))^{-1} $$

Match with the given equation form

$$ 2ab^{-1} = 2\ln(5)(\ln(6))^{-1} \implies a = \ln(5),\ b = \ln(6) $$

Answer:

• (A) \(a = \ln(6)\) and \(b = \ln(5)\)
• (B) \(a = \ln(5)\) and \(b = \ln(6)\) (Correct answer)
• (C) \(a = \ln(25)\) and \(b = \ln(6)\)
• (D) \(a = \ln(5)\) and \(b = \frac{1}{\ln 6}\)