Basic Properties of the Exponential Function

Let $\exp$ be the exponential function. Then:

1. $\exp(0)=1$ and $\exp(u+v)=\exp(u)\exp(v)$ for all $u,v\in\mathbb{R}$;
2. $\exp(u)>0$ for every $u\in\mathbb{R}$, and $\exp(-u)=1/\exp(u)$;
3. $\exp$ is differentiable at every point with $\exp'=\exp$, where the derivative is the one-dimensional one; consequently $\exp$ is a smooth map on $\mathbb{R}=\mathbb{R}^1$;
4. $\exp$ is strictly increasing, $\exp(u)\ge 1+u$ for $u\ge 0$, and $\exp(u)\to 0$ as $u\to-\infty$ in the sense that for every $\varepsilon>0$ there is $M$ with $\exp(u)<\varepsilon$ for all $u<-M$;
5. $\exp$ is a bijection from $\mathbb{R}$ onto $(0,\infty)$.