Possible definitions of exponential function

I was wondering how many definitions of exponential functions can we think of. The basic ones could be:

ex:=k=0xkk!
also
ex:=limn(1+xn)n
or this one:
Define ex:RR as unique function satisfying:
exx+1x,yR:ex+y=exey
Can anyone come up with something unusual? (Possibly with some explanation or references).

Answer

The exponential function is the unique solution of the initial value problem

y(x)=y(x),y(0)=1.

Attribution
Source : Link , Question Author : Community , Answer Author : Fred

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