Compare $2^{2016}$ and $10^{605}$ without a calculator So, I am supposed to compare $2^{2016}$ and $10^{605}$ without using a calculator, I have tried  division by $2$ on both sides  and then comparing $2^{1411}$ and $5^{605}$, and then substituting with $8,16,10$ and then raising to powers and trying to prove that but that did not go anywhere, I have also tried taking $\log$ of both sides, but did not help either. Also is there a more general approach to these kind of problems?

$\log_{10}2^{2016}=2016\log_{10}2\approx 606.88>605$
If calculators are not allowed, we have
\begin{align*}
2^{2016}&=64(1000+24)^{201}\\
&>64\left[1000^{201}+\binom{201}{1}1000^{200}(24)\right]\\
&=64(10^{603})(5.824)\\
&>100(10^{603})\\
&=10^{605}
\end{align*}
