Power tower inequality

Question

I want to prove the following power tower inequality:

$$ 3 \uparrow \uparrow 100 > 4 \uparrow \uparrow 99 $$

but I don't know how to do this. I think that induction will not work, because I think there will be an $N$ for which

$$ 3 \uparrow \uparrow N < 4 \uparrow \uparrow (N-1) $$

Could anyone help me in the right direction? Please don't answer a full solution, but I do need a hint.

Answer 1

Try to prove

$3 \uparrow \uparrow n+1 > \log_3(4) \cdot 4 \uparrow \uparrow n$ for all $n \in \mathbb{N}$

and use the following facts about $\log$

• $\log_a (x) = \frac{\log_b(x)}{\log_b(a)}$
• $\ln : \mathbb{R}_+ \rightarrow \mathbb R$ is strictly increasing