Complex exponents with positive real bases

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

$B^{\coprod_{j\in J}X_j}\cong\prod_{j\in J} B^{X_j}$