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  4. The value of ( log 3 12/ log 36 3)-( log 3 4/ log 108 3) is equal to

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Q. The value of $\frac{\log _3 12}{\log _{36} 3}-\frac{\log _3 4}{\log _{108} 3}$ is equal to

Continuity and Differentiability

Solution:

$ \log _3 12 \cdot \log _3 36-\log _3 4 \cdot \log _3 108=2\left(2 \log _3 2+1\right)\left(\log _3 2+1\right)-\left(2 \log _3 2\right)\left(3+2 \log _3 2\right)$
Let $\log _3 2= t$
$\therefore 2(2 t+1)(t+1)-2 t(2 t+3)=2\left(2 t^2+3 t+1-2 t^2-3 t\right)=2 \times 1=2$.