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

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

$f:ℝ⟶ℝ,x⟼b^x,$

fulfills the following properties.

1. We have $b^{x+x^{\prime }}=b^x\cdot b^{x^{\prime }}$ for all $x,x^{\prime }\in ℝ$.
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 $(b^x{)}^{x^{\prime }}=b^{x\cdot x^{\prime }}$ for all $x,x^{\prime }\in ℝ$.
8. For $a\in {ℝ}_{+}$, we have $(ab{)}^x=a^x\cdot b^x$.

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