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?
$$2^{2016}=(2^{10})^{201}\cdot2^6=1024^{201}\cdot64$$

$$10^{605}=(10^3)^{201}\cdot10^2=1000^{201}\cdot100$$

Hence by Bernoulli's Inequality, $$\frac{2^{2016}}{10^{605}}=\left(\frac{1024}{1000}\right)^{201}\cdot\frac{64}{100}=1.024^{201}\cdot0.64&gt;(1+201\cdot0.024)\cdot0.64&gt;5.8\cdot0.64&gt;1$$ so $$\boxed{2^{2016}&gt;10^{605}}$$
\begin{aligned}
2 ^ {2016} &gt; 10 ^ {605} &amp; \iff \\
e ^ {\ln (2 ^ {2016}) } &gt; e ^ {\ln (10 ^ {605}) } &amp; \iff \\
\ln (2 ^ {2016}) &gt; \ln (10 ^ {605}) &amp; \iff \\
2016 \cdot \ln (2) &gt; 605 \cdot \ln (10) &amp; \iff \\
\frac{2016}{605} &gt; \frac{\ln 10}{\ln 2} &amp; \iff \\
\frac{2016}{605} &gt; \frac{\log 10}{\log 2} &amp; \iff \\
\frac{2016}{605} &gt; \frac{1}{\log 2} &amp; \iff \\
3.33 \ldots &gt; \frac{1}{0.301 \ldots} &amp; \iff \\
3.33 \ldots &gt; 3.32 \ldots
\end{aligned}

PS: the two boxed expressions (i.e $\boxed{ \frac{2016}{605} }$ and $\boxed{ \log 2}$ ) can be derived by hand: one can divide on paper and (as far as I can remember) during tests you can bring logarithmic tables