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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?

3nondatur
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    What could be simpler? You use the two input values $n,m$ once each, you apply a simple standard function to each, you combine the two by the simple operation of multiplication – Hagen von Eitzen Nov 17 '18 at 11:21

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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.

Matt Samuel
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