Question

If ${\log }_{10}7=0.8451$ and ${\log }_{10}11=1.0414$, then the number of digits in ${77}^{100}$ is

• A. $187$
• B. $188$
• C. $189$
• D. $190$

Answer 1

The correct option is C $189$

Let $N={77}^{100}$
Then ${\log }_{10}N={\log }_{10}{77}^{100}$
$\Rightarrow {\log }_{10}N=100{\log }_{10}(7\times 11)$
$\Rightarrow {\log }_{10}N=100({\log }_{10}7+{\log }_{10}11)$
$\Rightarrow {\log }_{10}N=100\times (0.8451+1.0414)$
$\Rightarrow {\log }_{10}N=100\times 1.8865=188.65$

$\therefore$ Characteristic of ${\log }_{10}N$ is $188$.
Hence, the number of digits in ${77}^{100}$ is $189$.