Let $A,B$ be full rank $n \times n$ matrices. If $AB = BA$, then $\exp(\log(A)+\log(B))=AB$.

Supposing $A = USL$ and $B = VSL$ where $U,V,S,L$ are integer valued matrices, $det(L)=1$ and $U = LVL^{-1}$. If $AB = (USL)(VSL) = (L(VSL)L^{-1})(L^{-1}(USL)L) = (LBL^{-1})(L^{-1}AL)$, is there a possibility to use exponentials to calculate $AB$?