If logaba=4, then the value of logab[a3b] is

logaba=4logaab=14 1+logab=14 logab=−34 logabblogaba=−34 logabb4=−34 logabb=−3 logab[a3b]=13logaba−12logabb 13(4)−12(−3)=176

If logaba=4, then the value of logab[a3b] is

logaba=4logaab=14 1+logab=14 logab=−34 logabblogaba=−34 logabb4=−34 logabb=−3 logab[a3b]=13logaba−12logabb 13(4)−12(−3)=176

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If $\log_{a b} a = 4$ , then the value of $\log_{a b} [\frac{\sqrt[3]{a}}{\sqrt{b}}]$ is

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$\frac{17}{6}$

$\frac{6}{17}$

$\frac{7}{6}$

$\frac{6}{7}$

answer is A.

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Detailed Solution

$\log_{a b} a = 4$

$\log_{a} a b = \frac{1}{4} 1 + \log_{a} b = \frac{1}{4} \log_{a} b = \frac{- 3}{4} \frac{\log_{a b} b}{\log_{a b} a} = \frac{- 3}{4} \frac{\log_{a b} b}{4} = \frac{- 3}{4} \log_{a b} b = - 3 \log_{a b} [\frac{\sqrt[3]{a}}{\sqrt{b}}] = \frac{1}{3} \log_{a b} a - \frac{1}{2} \log_{a b} b \frac{1}{3} (4) - \frac{1}{2} (- 3) = \frac{17}{6}$