Compare power towers

Question

Prove or disprove:
$3^{3^{3^{3^{3...^3}}}}$ with 100 threes $>4^{4^{4^{4^{4...^4}}}}$ with 99 fours.

Taking logs is useless, and there seems to be no other way to compare. Thanks!

Answer 1

It suffices to see that $3^3>6\times4$ and that

$$3^{6n}>6\times4^n$$

for all $n\ge1$. By induction this gives us:

$$3\uparrow\uparrow(n+1)>6(4\uparrow\uparrow n)$$

for all $n\ge1$.

Of course much better bounds can be given, but this suffices.