Question

${\log }_7{\log }_7\sqrt{7\sqrt{7\sqrt{7}}}$ is equal to

• A. $3{\log }_{2}7$
• B. $1-3{\log }_{3}7$
• C. $1-3{\log }_{7}2$
• D. None of these

Answer 1

The correct option is C $1-3{\log }_{7}2$

Let $a={\log }_7{\log }_7\sqrt{7\sqrt{7\sqrt{7}}}$

$={\log }_7{\log }_7\sqrt{7\sqrt{7\times 7^{\frac{1}{2}}}}$

$={\log }_7{\log }_7\sqrt{7\times {\left(7^{\frac{3}{2}+1}\right)}^{\frac{1}{2}}}$

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

$={\log }_7{\log }_7{\left(7^{1+\frac{3}{4}}\right)}^{\frac{1}{2}}$$={\log }_7{\log }_7{7}^{7/8}$

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

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

$=1-{\log }_{7}8=1-{\log }_7{2}^3$

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

$\therefore {\log }_7{\log }_7\sqrt{7\sqrt{7\sqrt{7}}}=1-3{\log }_{7}2$