Question

The number of digits in 20^301 ( log 2to the base 10 = 0.30103

Answer 1

The number of digits in a number $a^b$ is given by $\lceil\log_{10}\left (a^b\right)\rceil,$ where $\lceil x\rceil$ represents the smallest integer greater than but not equal to $x.$

So, to the base 10,

$\lceil\log\left (20^{301}\right)\rceil=\lceil 301\log (20)\rceil\\\\\\=\lceil 301\log(2\times 10)\rceil=\lceil301\left (\log 2+\log 10\right)\rceil\\\\\\=\lceil 301(0.30103+1)\rceil=\lceil301\times 1.30103\rceil\\\\\\\lceil391.61003\rceil=\mathbf {392}$