Question

Find the number of digits in 4²⁰¹³. If_{​​​​​} log_​​​10​​2 = 0.3010

Answer 1

Given, $4^{2013}$

Let us take

$log\left(4^{2013}\right)$

$=log{\left[{\left(2\right)}^2\right]}^{2013}-----[\because {\left(a^m\right)}^n=a^{mn}]$

$=log{\left(2\right)}^{4026}$

$=4026log\left(2\right)-----[\because log{m}^n=nlogm]$

$=4026\times 0.301-----[\because log2=0.301]$

$=1211.826$

$Thus{4}^{2013}has4digitsbeforethedecimals$