Question

Prove that: ${\log }_7{\log }_7\sqrt{7\sqrt{\left(7\sqrt{7}\right)}}=1-3{\log }_{7}2$

Answer 1

Let $L={\log }_7{\log }_7\sqrt{7\left(\sqrt{7\sqrt{7}}\right)}$

$={\log }_7{\log }_7\sqrt{7\left(\sqrt{7^{3/2}}\right)}$

$={\log }_7{\log }_7\sqrt{7\left(7^{3/4}\right)}$

$={\log }_7{\log }_7\sqrt{7^{7/4}}$

$={\log }_7{\log }_7{7}^{7/8}$

$={\log }_7\left(\frac{7}{8}{\log }_{7}7\right)$

$={\log }_7\left(\frac{7}{8}\right),$ ($\because {\log }_{a}a=1$)

$={\log }_{7}7-{\log }_{7}8$

$=1-{\log }_7{2}^3,$ ($\because {\log }_{a}a=1$)

$L=1-3{\log }_{7}2$

${\log }_7{\log }_7\sqrt{7\left(\sqrt{7\sqrt{7}}\right)}=1-3{\log }_{7}2$

Hence, proved