Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$?

Question

It is the first time I met such a question:

Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$?

Intuitively I think $f(n)$ would gradually become larger as $n$ gets larger, but I find it hard to produce an argument. Is there any trick to use for this type of question?

Answer 1

Compare $f(n)$ with $G(n)=256^{256^n}>g(n)$. You arrive at $G(n) = 2^{2^{8n+3}}$, so you just need to compare $2^n$ vs. $8n+3$.

Answer 2

Try taking the logs. Log is a monotonic transformation.

Answer 3

Take logs

$$\log\log\log(f)=\log\log\log(2^{2^{2^n}})=n\log(2\log(2\log(2)))$$ $$\log\log(g)=\log\log(100^{100^n})=n\log (100(\log (100)))$$

Which are both linear and grow at the same rate, so to say. Now exponentiate to retrieve $f,g$.