Can the asymptotic expression $\mathcal{O(log(n) \cdot log(m))}$ be simplified?

Question

I constructed an algorithm $A$ with input $(n,m)$ and I found that it has runtime $\mathcal{O(log(n) \cdot log(m))}$. I was asking myself if this expression can be simplified somehow, but I could not find a way. Do you see a possiblity here?

Answer 1

The only thing I can think of is $$\mathcal O(\log(n^{\log m})) $$ But this is silly and not really any simpler. Go with what you have.