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Two recent questions have asserted in comments that one may define $\exp x$ as the unique function on $\mathbb{R}$ for which we have $\exp'=\exp$ and $\exp 0=1$. Can one see directly that such a function exists? Or must one either (a) define the usual power series, prove it is convergent and differentiable everywhere, and so is a candidate for $\exp x$; or alternatively (b) define $\log x:=\int_{1}^{x}\frac{dt}{t}$ on $\mathbb{R}^{>0}$, prove enough about it to show that it has an inverse, which by the FTC is a candidate for $\exp$.

This related question Exponential Function - Definition seems to me to consist of mere assertions, but it does point one to Wikipedia: Characterizations of the Exponential Function, where the above "definition" is Characterization 4.

Joe
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ancient mathematician
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    That really depends on what you can use. If you know some basic theory about differential equations then you can “see directly” that a unique solution to the initial value problem $y' = y$, $y(0) = 1$ exists. – Martin R Feb 15 '21 at 15:50
  • In the normal course of mathematical learning, and if you want to be rigorous, I would expect that the existence of a unique solution to the differential equation would come later than the desirability of having the exponential function available, so to me better to define $\exp x$ by some other route than as the unique solution to $y'=y, y(0) = 0$. – WA Don Feb 15 '21 at 15:56
  • @WADon: In my book a mature math learner would have a couple of different definitions of $\exp$ available, and have seen proofs of their equivalence. So, should that be the first definition they see? Probably not. Should they know it? Yes. – JonathanZ Feb 15 '21 at 16:01
  • @MartinR I think I am asking for an account of this "basic theory about DE": is there some big theorem I seem to have missed which asserts the existence of a solution of certain sorts of DE? – ancient mathematician Feb 15 '21 at 16:02
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    @ancientmathematician The Picard–Lindelöf theorem is probably what you are looking for. – Aryaman Maithani Feb 15 '21 at 16:04
  • @AryamanMaithani Thanks, I knew that if there was such an account it would have to be basically a fixed-point theorem. But this seems to me to pretty deep stuff to learn before defining $\exp$. ;-) – ancient mathematician Feb 15 '21 at 16:09
  • @ancientmathematician Alternatively, one could use the Peano existence theorem. This theorem only proves that the differential equation at hand has solutions (not that the solution is unique). However, in the case of the exponential function, once you have proved existence, uniqueness can be proven using elementary calculus (see here). – Joe Feb 15 '21 at 17:10
  • In addition, this thread about the different ways of introducing the elementary functions might be of use to you. – Joe Feb 15 '21 at 17:10

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