Question

The value of ${\log }_7{\log }_7\sqrt{7\sqrt{7\sqrt{7}}}$ is

• A. $3{\log }_{2}7$
• B. $3{\log }_{2}2$
• C. $1-3{\log }_{7}2$
• D. $1-3{\log }_{2}7$

Answer 1

Answer: (C)

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

We have,

$\begin{array}{c}{\log }_7\log 7\sqrt{7\sqrt{7\sqrt{7}}}is\\ \Rightarrow {\log }_7\left[{\log }_7\left(\sqrt{7\sqrt{7\sqrt{7}}}\right)\right]\\ \Rightarrow {\log }_7\left[\log {\left(7\sqrt{7\sqrt{7}}\right)}^{\frac{1}{2}}\right]\left[\log a^n=n\log a\right]\\ \Rightarrow {\log }_7\left(\frac{1}{2}\log \left(7\sqrt{7\sqrt{7}}\right)\right)\\ \Rightarrow {\log }_7\left(\frac{1}{2}{\log }_7\left({7.7}^{3/4}\right)\right)\\ \Rightarrow {\log }_7\left(\frac{7}{8}\right)={\log }_{7}7-{\log }_{7}8=\left(1-3{\log }_{7}2\right)\end{array}$

Hence, this is the answer.