Question

Given that ${\log }_{10}2=0.3010$, the number of digits in the number ${2000}^{2000}$ is

• A. $6600$
• B. $6603$
• C. $6543$
• D. $2345$

Answer 1

The correct option is B $6603$

Let $x={2000}^{2000}$
$\Rightarrow {\log }_{10}x=2000{\log }_{10}2000$
$\Rightarrow {\log }_{10}x=2000{\log }_{10}(2\times 1000)$
$\Rightarrow {\log }_{10}x=2000({\log }_{10}2+{\log }_{10}1000)$
$\Rightarrow {\log }_{10}x=2000(0.3010+3)$
$\Rightarrow {\log }_{10}x=2000\left(3.3010\right)=6602$

$\therefore \text{Number of digits}=6603$