Which of the numbers $300!$ and $100^{300}$ is greater Determine which of the two numbers $300!$ and $100^{300}$ is greater.
My attempt:Since numbers starting from $100$ to $300$ are all greater than $100$. But am not able to justify for numbers between $1$ to $100$

Consider $$k! > n^k$$ and note that $$\lim_{k\to \infty} \frac{(k!)^{1/k}}{k} = \frac{1}{e}$$ 
This implies that if $n$ is smaller than $\frac{k}{e}$, the first statement is true. So if we take $n = 100$ and $k = 300$, we get
$$100 < \frac{300}{e}$$
and $$300! > 100^{300}$$
