| context | $ b\in\mathbb R_+^* $ |
| definiendum | $ z\mapsto b^z :\mathbb C\to\mathbb C $ |
| definiendum | $ z\mapsto b^z := \mathrm{exp}(z\cdot \mathrm{ln}(b)) $ |
The identity
$b^{x_1+x_2}=b^{x_1}\cdot a^{x_2}$,
says that exponentiation is a (the) homomorphism between $+$ and $\cdot$.
The combinatorial manifestation, e.g. formulated in for $B,X_1,X_2,\dots\in\bf{Set}$, is
$B^{\coprod_{j\in J}X_j}\cong\prod_{j\in J} B^{X_j}$