Assume that ${\mathcal{X}}$ is snc, and write as above ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$. Pick a closed point $\xi\in{\mathcal{X}}_{0}$, and denote by $J=\{0,\dots,p\}\subset I$ the set of components of ${\mathcal{X}}_{0}$ passing through $\xi$. We may choose a regular system of parameters $z_{0},\dots,z_{n}\in{\mathcal{O}}_{{\mathcal{X}},\xi}$ such that $z_{i}$ is a local equation of $E_{i}$ for $0\leq i\leq p$, i.e. ${t}=uz_{0}^{b_{0}}\dots z_{p}^{b_{p}}$ for some unit $u\in{\mathcal{O}}^{*}_{{\mathcal{X}},\xi}$. The logarithmic form

$$\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}$$

is then a local generator of $K^{\mathrm{log}}_{{\mathcal{X}}}$, and induces a local generator

$$\Omega^{\mathrm{rel}}:=\Omega\otimes(\frac{d{t}}{{t}})^{-1}$$

of $K^{\mathrm{log}}_{{\mathcal{X}}/S}$.