Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$? 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?
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 $f,g$.
Try taking the logs. Log is a monotonic transformation.