Do factorials really grow faster than exponential functions? Having trouble understanding this. Is there anyway to prove it?
If you're not quite in the market for a full proof:

$$a^n=a\times a\times a\times a...\times a$$
$$n!=1\times 2\times 3\times 4...\times n$$

Now what happens as $n$ gets much bigger than $a$? In this case, when $n$ is huge, $a$ will have been near some number pretty early in the factorial sequence. The exponential sequence is still being multiplied by that (relatively tiny) number at each step, while $n!$ is being multiplied by $n$. So even if $n!$ starts out small, it'll eventually start being multiplied by gigantic numbers at each step, and quickly outgrow the exponential. If $a=10$ and $n=100$, then $a^n$ has around $100$ digits, while $n!$ has over $150$ digits. Note that near $n=100$, $n!$ is having roughly 2 digits added per step (and that rate will only increase), while $a^n$ is still only ever going to get one more with every step. No contest.
$n!&gt; k^n$ if $n\ge ke$

try it for yourself with various combinations of $n$ and $k$

use $\log(n!)$ and $k \log(n)$ for large $n$