> the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. But i

> the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. But it works when "this" is an action that does not have a sensible notion of being applied a fractional number of times, or an infinite number of times, which is a very large and important set of actions indeed. > When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge Yes of course, but mere continuity of the inputs doesn't pick out exp from any other base you might choose besides e. You do not have to relate it to the exp function. > Since the OP was about a notation propos... My specific objection was to your suggestion that a^b is syntactic sugar, which suggests notational convenience that does not reflect what's going on "under the hood".