Let $\exp$ be the \reftext{def:exponential-function-real-2026a}{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 \reftext{def:derivative-interior-point-c54-2026b}{derivative} is the one-dimensional one; consequently $\exp$ is a \reftext{def:smooth-map-euclidean-open-set-2026a}{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 \reftext{def:bijection-sets-2026a}{bijection} from $\mathbb{R}$ onto $(0,\infty)$.