There exists an open neighborhood ${\mathcal{V}}\subset{\mathcal{X}}$ of $D$ and a continuous map $\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}\setminus D\to\Delta(D)$ such that for each adapted coordinate chart $({\mathcal{U}},z)$ with ${\mathcal{U}}\subset{\mathcal{V}}$ we have $\operatorname{Log}_{\mathcal{V}}({\mathcal{U}}\setminus D)\subset\sigma_{\mathcal{U}}$ and

$$\operatorname{Log}_{\mathcal{V}}=\operatorname{Log}_{{\mathcal{U}}}+O\left(\frac{1}{\log|f_{{\mathcal{U}},z}|^{-1}}\right)$$

(2.2)

uniformly on compact subsets of ${\mathcal{U}}$.