In this video by "Three Blue One Brown" dealing with group theory https://www.youtube.com/watch?v=mvmuCPvRoWQ
the operation of exponentiation is presented as a group morphism, that is a structure preserving function from an additive group to a multplicative group.
the quick justification provided at 16:52 is the exponent law $n^{a+b} = n^a\times n^b$. I think the author only takes this law as an example of a more general " exponential property" that shows up regularly in group theory.
Is there a not too complicated way to make this idea more precise?
The homomorphism property is $h(x+y)=h(x)+h(y)$. The exponential map $h:(\Bbb R,+)\to(\Bbb R^+,\cdot)$, given by $h(x)=e^x$, indeed satisfies this property. We have $h(x+y)=e^{x+y}=e^x\cdot e^y=h(x)\cdot h(y)$.