Tricky one

Algebra Level 3

What is the least value of $n$ , such that ${ 2 }^{ n }$ contains exactly one million digits?

2 solutions

Abdulrahman El Shafei
Dec 17, 2014

$(n\times \log { 2) } +1=1000000$

$(n\times \log { 2) } =999999$

$n=\frac { 999999 }{ \log { 2 } } \approx \boxed { 3321924.773 }$

J Joseph
Nov 1, 2016

$2^{n} \geq 10^{1000000}$

$log_2 ({2^n}) \geq log_2 (10^{10^{6}})$

$n \geq log_2 ([2^{10^{6}}][5^{10^{6}}])$

$n \geq log_2(2^{10^{6}}) + log_2(5^{10^{6}})$

$n \geq 10^{6} + 10^{6}log_2(5)$