How to show that $2^n > n$ without induction I'm solving exercises about Pascal's triangle and Binomial theorem, and this problem showed up, however I don't have any clue on how to solve it
The sum of ${n\choose p}$ from $p=0$ to $n$ is the same thing as $(1+1)^n=2^n$, how can I use this information?
Maybe comparing with another summation that equals to n?

hint
Consider $x\mapsto \frac{\ln(x)}{x}$ for $x\ge 1$.
$$f'(x)=\frac{1-\ln(x)}{x^2}$$
the maximum if $f(e)=\frac{1}{2}<\ln(2)$.
thus
$$\ln(x)<x\ln(2)$$
