TITLE: Which of the numbers $300!$ and $100^{300}$ is greater
QUESTION [14 upvotes]: 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 $100$

REPLY [10 votes]: $\displaystyle e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!}+\frac{x^{n+1}}{(n+1)!}+\cdots $
$$e^x>\frac{x^n}{n!}$$  for $x=n$
$$n!>\bigg(\frac{n}{e}\bigg)^n>\bigg(\frac{n}{3}\bigg)^n$$
for $n=300.$  we have $\color{red}{300!>(100)^{300}}$