Question

Find the value of ${\log }_7{\log }_7$ $\sqrt{7\left(\sqrt{7\sqrt{7}}\right)}$ , if ${\log }_{10}$ 7 = 0.8450 and ${\log }_{10}$ 2 = 0.3010

• A. 0.02000
• B. 0.4225
• C. 0.0685
• D. 1.2676

Answer 1

The correct option is C

0.0685

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

= ${\log }_7{\log }_7$ $\sqrt{7.\sqrt{7.\sqrt{7^{\frac{1}{2}}}}}$ = ${\log }_7{\log }_7$ $\sqrt{7^{\frac{7}{4}}}$

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

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

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

= 1 - 3$\frac{{\log }_{10}2}{{\log }_{10}7}$ = 1 - 3 $\times$ $\frac{0.3010}{0.8450}$ = 1 - 3 $\times$ 0.3561

= 0.0685.