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  4. The value of N =2 log 63 3(1+3 log 32 2)- log 63((3/2)) is equal to

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Q. The value of $N =2 \log _6^3 3\left(1+3 \log _3^2 2\right)-\log _6^3\left(\frac{3}{2}\right)$ is equal to

Continuity and Differentiability

Solution:

$ \frac{2}{\left(\log _3 6\right)^3}\left(1+3 \log _3^2 2\right)-\frac{\left(1-\log _3 2\right)^3}{\left(1+\log _3 2\right)^3}$
Let $\log _3 2= t$
$=\frac{2\left(1+3 t^2\right)}{(1+t)^3}-\frac{(1-t)^3}{(1+t)^3} $
$=\frac{2+6 t^2-1+t^3+3 t-3 t^2}{(1+t)^3} $
$=\frac{t^3+3 t+3 t^2+1}{(1+t)^3}=1 $