TITLE: Definition of $\exp(x)$
QUESTION [5 upvotes]: I've been thought at school that the definition of $\exp(x)$ is the unique function satisfying $\exp'(x)=\exp(x)$ and $\exp(0)=1$. But I've never found this definition somewhere else. On Wikipedia there are multiple definitions which are :
$$\exp(x)=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$
$$\text{The solution $y$ to the equation }x=\int_1^y\frac{1}tdt$$
$$\text{and } \exp(x)=\lim_{n\to+\infty}\left(1+\frac{x}{n}\right)^n$$
As I know, something can have only one definition, but can have multiply ways to define it and properties. So which one is the real definition of the exponential function?

REPLY [2 votes]: Another Wikipedia article of relevance is Characterizations of the exponential function.
Which characterization is most appropriate to take to be the definition depends on the context.