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

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

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

fulfills the following properties.

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

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