Matrix exponentiation of a matrix

Question

I search a definition of the exponential $A^B$ where $A$ and $B$ are $n\times n$ matrices. I suppose that, if $A$ is invertible so that there exists a matrix $log(A)$ that is the (principal) logaritm of $A$ ( i.e. $e^{log(A)}=A$), then we can define $A^B=e^{(log A)B}$, but in general this is different from $A^B=e^{B(log A)}$. So, what is (if it exist) the accepted definition of $A^B$ ?

Answer 1

$A = e^{logA}$ so $$A^B = (e^{logA})^B$$ We would multiply the B from left to right so $$A^B = e^{(logA)B}$$