# When can one use logarithms to multiply matrices

If $a,b \in \mathbb{Z}_{+}$, then $\exp(\log(a)+\log(b))=ab$. If $A$ and $B$ are square matrices, when can we multiply $A$ and $B$ using logarithms? If $A \neq B^{-1}$, should $A$ and $B$ be symmetric?

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If $A$ and $B$ commute, i.e. $AB=BA$, then the same identity holds for an appropriate definition of $\exp$ and $\log$. If $A$ and $B$ do not commute, then things are much more complicated.
When they commute, or: when they have the same eigenvectors, or if $AB=BA$