Exponential of Zero and One

Theorem
Let $\exp x$ be the exponential of $x$.

Then:
 * $\exp 0 = 1$
 * $\exp 1 = e$

where $e$ is Euler's number, i.e. $2.718281828\ldots$.

Proof 1
We have that the exponential function is the inverse of the natural logarithm function:


 * $\ln 1 = 0$
 * $\ln e = 1$

Hence the result.

Proof 2
We use the definition of the exponential as a limit of a sequence:

That $\exp 1 = e$ follows from the Equivalence of Definitions of Euler's Number.