Pick an open cover $({\mathcal{V}}_{\alpha})_{\alpha}$ as in Lemma 2.2, and denote by $\operatorname{Log}_{\alpha}\colon{\mathcal{V}}_{\alpha}\setminus D\to\sigma_{\alpha}$ the corresponding maps. Set ${\mathcal{V}}:=\bigcup_{\alpha}{\mathcal{V}}_{\alpha}$, and pick a partition of unity $(\chi_{\alpha})$ subordinate to $({\mathcal{V}}_{\alpha})$. We claim that for each $\xi\in{\mathcal{V}}$ there exists an open neighborhood $W$ of $\xi$ and a face $\sigma_{W}$ of $\Delta(D)$ such that

$$W\cap\operatorname{supp}\chi_{\alpha}\neq\emptyset\Longrightarrow\sigma_{\alpha}\subset\sigma_{W}$$

for any $\alpha\in A$. Indeed, using (2.3) it is easy to see that

$$W:=\bigcap_{\alpha\mid\xi\in{\mathcal{U}}_{\alpha}}{\mathcal{V}}_{\alpha}\setminus\bigcup_{\alpha\mid\xi\notin\operatorname{supp}\chi_{\beta}}\operatorname{supp}\chi_{\beta}$$

satisfies this property. By convexity of $\sigma_{W}$, it follows that $\operatorname{Log}_{\mathcal{V}}:=\sum_{\alpha}\chi_{\alpha}\operatorname{Log}_{{\mathcal{V}}_{\alpha}}$ is well-defined on $W\setminus D$, and hence yields a continuous map $\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}\setminus D\to\Delta(D)$. The last property is a direct consequence of (2.1). ∎