Master Theorem linearithmic function

Question

I am trying to find the running time of the given recurrence by the Master Theorem: $T(n)=16T(\frac{n}{2})+n^3\log^4 n$

I get $a=16$, $b=2$ and $f(n)=n^3\log^4n$, It seems that it's Case 1 of the master theorem. because $f(n)=O(n^{4-\epsilon})$ for some $\epsilon >0$ (?)

but I can't find this epsilon, and I am doubting if I can truly use the Master Theorem in this case.

What's I am missing?

Answer 1

It's easy to prove that

$\qquad\displaystyle n^3 \log^k n \in o(n^{3+x})$

for all $k \geq 0$ and $x > 0$. That should give you plenty of $\epsilon$ to choose from.