Question

If $N =\operatorname{antilog}_7\left(\log _3\left(\operatorname{antilog}_{\sqrt{3}}\left(\log _3 81\right)\right)\right)$, then $\log _3 N$ lies between two successive integers $a$ and $b$. Find $(a+b)$.

Answer 1

$ N =\operatorname{antilog}_7\left(\log _3\left(\operatorname{antilog}_{\sqrt{3}}\left(\log _3 81\right)\right)\right)=\operatorname{antilog}_7\left(\log _3(\sqrt{3})^4\right) $
$=\operatorname{antilog}_7\left(\log _3 9\right)=\operatorname{antilog}_7(2)=7^2=49 $
$ N =49 \Rightarrow 3<\log _3 43<4 \Rightarrow a + b =7$