Power function in non-commutative algebras

Question

Is there a canonical form of the power function, $a^b=x$, that extends to non-commutative algebras like matrices and hypercomplex numbers? It is known that $a^b=e^{b\log{a}}$ for commutative algebras, but in commutative algebras, $e^{b\log{a}}=e^{\log(a)b}=(e^{\log{a}})^b$, which is not true when you don't have the luxury of commutativity. In other words, is there a standard way to define things like $M_1^{M_2}$ for $M_1,M_2 \in M_{n\times n}$ or $Q_1^{Q_2}$ for $Q_1, Q_2 \in \mathbb{H}$?