Let $T:={\mathbb{G}}_{m}^{r}$ be a split multiplicative torus over $K$ with coordinates $z_{1},\dots,z_{r}$. Then we have the tropicalization map

$${{\rm trop}}:{T^{\rm an}}\rightarrow{\mathbb{R}}^{r},\quad p\mapsto(-\log p(z_{1}),\dots,-\log p(z_{r})).$$

It is immediate from the definitions that the map ${{\rm trop}}$ is continuous and proper. To get a coordinate free approach, we could use the character group $M$ and its dual $N$. Then ${\rm trop}$ is a map from ${T^{\rm an}}$ to $N_{\mathbb{R}}$. We refer to [Gu12] for details about tropical geometry.