Does $(\frac{n}{4})^{\frac{n}{4}}$ have a higher asymptotic growth than $4^{n^{4}}$

Question

I'm trying to determine how these 3 functions should be ordered in terms of asymptotic growth: $$f(n) = \left(\frac{n}{4}\right)^{\frac{n}{4}}$$ $$g(n) = n^{\frac{n}{4}}$$ $$h(n) = 4^{n^{4}}$$

$f(n)$ seems to be somewhat similar to $n^{n}$ and that should grow faster than $4^{n}$, but I'm not sure how to prove it.

Answer 1

Taking logs on both sides you have the comparison

$$\log g = \frac{n}{4}\log n$$

$$\log h = n^4 \log 4$$

Which is faster?