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  4. The value of ( log 5 9 ⋅ log 7 5 ⋅ log 3 7/ log 3 √6)+(1/ log 4 √6) is co-prime with

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Q. The value of $\frac{\log _5 9 \cdot \log _7 5 \cdot \log _3 7}{\log _3 \sqrt{6}}+\frac{1}{\log _4 \sqrt{6}}$ is co-prime with

Continuity and Differentiability

Solution:

$\frac{\log _5 9 \cdot \log _7 5 \cdot \log _3 7}{\log _3 \sqrt{6}}+\frac{1}{\log _4 \sqrt{6}}$
$\frac{\log _3 9}{\frac{1}{2} \log _3 6}+\frac{1}{\frac{1}{2} \log _4 6} $
$\frac{2 \cdot 2}{\log _3 6}+\frac{2}{\log _4 6} $
$4 \cdot \log _6 3+2 \log _6 4 \Rightarrow \log _6 3^4+\log _6 4^2 $
$=\log _6\left(3^4 \cdot 2^4\right)=4$