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:R→R as unique function satisfying:

ex≥x+1∀x,y∈R: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*