Exponential series/Real/Elementary properties/Fact

The exponential function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \exp x,$

fulfills the following properties.

1. ${}\exp 0=1$.
2. For every ${}x\in \mathbb {R}$, we have ${}\exp {\left(-x\right)}=(\exp x)^{-1}$. In particular ${}\exp x\neq 0$.
3. For integers ${}n\in \mathbb {Z}$, the relation ${}\exp n=(\exp 1)^{n}$ holds.
4. For every ${}x$, we have ${}\exp x\in \mathbb {R} _{+}$.
5. For ${}x>0$ we have ${}\exp x>1$, and for ${}x<0$ we have ${}\exp x<1$.
6. The real exponential function is strictly increasing.

Proof, Write another proof