I was plugging in huge values for n and noticed n^0.01 grows incredibly slowly, much slower then log(n) it seems. So my initial thought was that $$n^{0.01}~is~BigO(log(n))$$.

However I came across the rule in my book that,

$$log^x(n)~is~BigO(n^y)$$ for any fixed constant x > 0 and y > 0,

which made me think my initial thought was wrong.