# Real exponential function/Base/Properties/Fact/Proof/Exercise

Let ${\displaystyle {}b}$ denote a positive real number. Prove that the exponential function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto b^{x},}$

fulfills the following properties.

1. We have ${\displaystyle {}b^{x+x'}=b^{x}\cdot b^{x'}}$ for all ${\displaystyle {}x,x'\in \mathbb {R} }$.
2. We have ${\displaystyle {}b^{-x}={\frac {1}{b^{x}}}}$.
3. For ${\displaystyle {}b>1}$ and ${\displaystyle {}x>0}$, we have ${\displaystyle {}b^{x}>1}$.
4. For ${\displaystyle {}b<1}$ and ${\displaystyle {}x>0}$, we have ${\displaystyle {}b^{x}<1}$.
5. For ${\displaystyle {}b>1}$, the function ${\displaystyle {}f}$ is strictly increasing.
6. For ${\displaystyle {}b<1}$, the function ${\displaystyle {}f}$ is strictly decreasing.
7. We have ${\displaystyle {}(b^{x})^{x'}=b^{x\cdot x'}}$ for all ${\displaystyle {}x,x'\in \mathbb {R} }$.
8. For ${\displaystyle {}a\in \mathbb {R} _{+}}$, we have ${\displaystyle {}(ab)^{x}=a^{x}\cdot b^{x}}$.