Question

The value of ${\log }_{\sqrt{2}}\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}},$ is

Answer 1

${\log }_{\sqrt{2}}\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}$

$=\frac{2}{2}{\log }_{2}2\sqrt{2\sqrt{2\sqrt{2}}}$

$={\log }_{2}2\sqrt{2\sqrt{2\sqrt{2}}}$

$={\log }_{2}2+{\log }_2\sqrt{2\sqrt{2\sqrt{2}}}$

$=1+\frac{1}{2}{\log }_{2}2\sqrt{2\sqrt{2}}$

$=1+\frac{1}{2}[{\log }_{2}2+{\log }_2\sqrt{2\sqrt{2}}]$

$=1+\frac{1}{2}+\frac{1}{2}\left[\frac{1}{2}{\log }_{2}2\sqrt{2}\right]$

$=\frac{3}{2}+\frac{1}{2}\left[\frac{1}{2}({\log }_{2}2+{\log }_2\sqrt{2})\right]$

$=\frac{3}{2}+\frac{1}{2}\left[\frac{1}{2}(1+\frac{1}{2}{\log }_{2}2)\right]$

$=\frac{3}{2}+\frac{1}{2}\times \frac{1}{2}(1+\frac{1}{2})$

$=\frac{3}{2}+\frac{1}{4}\times \frac{3}{2}$

$=\frac{3}{2}\times \frac{5}{4}$

$=\frac{15}{8}$