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$The value of \frac{\log_{3} 135}{\log_{15} 3} - \frac{\log_{3} 5}{\log_{405} 3} is$

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Detailed Solution

$\frac{\log_{3} 135}{\log_{15} 3} - \frac{\log_{3} 5}{\log_{405} 3}$

$\begin{array}{l} = (3 + \log_{3} 5) (1 + \log_{3} 5) - \log_{3} 5 \log_{3} 405 \\ = (3 + \log_{3} 5) (1 + \log_{3} 5) - \log_{3} 5 \log_{3} (81 \times 5) \\ = (3 + \log_{3} 5) (1 + \log_{3} 5) - \log_{3} 5 (4 + \log_{3} 5) = 3 \end{array}$

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