Function Growth Question

Question

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

I tried differentiating the terms but it didn't really reveal anything. Can anyone come up with a solution?

Thanks in advance for any contributions.

Answer 1

Hint: try comparing $$\ln \ln f(n) \quad \text{vs} \quad \ln \ln g(n).$$ That works in all similar cases.

Answer 2

To anyone still left, is this sufficient proof?

$\lim_{n \to \infty} \frac{2^{2^{2^n}}}{100^{100^n}} \\=\lim_{n \to \infty} \frac{ \ln \ln 2^{2^{2^n}}}{\ln \ln 100^{100^n}} \\=\lim_{n \to \infty} \frac{2^n \ln 2 + \ln \ln 2}{n \ln 100 + \ln \ln 100} \\=\lim_{n \to \infty} \frac{2^n}{n} \\= +\infty$