Is the growth on an exponential function faster than on a power function? Is this explanation valid?

Question

I'm trying to find simple mathematical proof that lengthy passwords are more secure than complex (with multiple symbols) ones. I don't want to get into the concept of entropy - it is not useful to me atm. I've found this explanation at Security Stack, which it seemed pretty reasonable to me. My question is: is this explanation valid and, if so, is there a limit to it's validity?. My doubt comes from testing it out considering a password with $n$ digits, but restrict to a group of 26 characters (the alphabet) vs a password with fixed 10 digits, with $n$ possible characters:

$$f(n) = 26^n \qquad\text{vs}\qquad f(n) = n^{10}.$$

For these two functions it just doesn't work out as the answer in Security Stack explained. So, how does this work?

I guess the title may be a bit misleading, because this question markes as duplicate does not answer my question! I get that exponential growth is faster than polynomial growth, but why doesn't it apply to these two functions I'm comparing? I mean, what is wrong in my line of thought?

Answer 1

It's certainly true that $a^x,\,a>1$ eventually grows faster than $x^b,\,b>0$. (Perhaps the easiest proof is the fact that $\int_0^\infty xe^{-cx}dx=\frac{1}{c^2},\,c>0$ is finite. Thus $\lim_{x\to\infty}xe^{-cx}=0$, so $x^b$ can't keep up with $e^{bcx}$; now take $c=b^{-1}\ln a$.) So $26^x>x^{10}$ for all sufficiently large $x$.

Answer 2

Look at what happens when $x$ increases from $1\text{ million}$ to $1\text{ million}+1.$

$3^x$ gets multiplied by $3.$ And does so every time $x$ increases by $1.$

But $x^{10}$ gets multiplied by something that's very small by comparison to $(1\text{ million})^{10}$ (even though it's very large in absolute terms).