Canonical log structure defined by a normal crossing divisor Given a locally noetherian scheme $X$ and $D \subset X$  a normal crossing divisor.
 Let $j: U=X-D \hookrightarrow X$ the open complement immersion. Define the log structure $(M_X,\alpha_X)$ on $X$ as the direct image by $j$ of the trivial log structure on $X$.
 Can someone explain to me why the morphism $\mathbb{N}_U^r \to M_U$ defined by $e_i \mapsto t_i$ where $t_i$ are part of a regular system of parameters of $\mathcal{O}_{X,x}$ and $e_i$ are basis of $\mathbb{N}^r$ is chart of $X$ on $U$; and especially the fact that at a geometric point $x$ of $U$ every element $a$ of $M_x$ can be uniquely written as $a=u \prod_{i \in I} t^{n_i}_i$ with $n_i \in \mathbb{N}$, $u \in \mathcal{O}_x^*$, and $I$ is the set of $I$ such that $(t_i)_x \in \mathfrak{m}_x$?
Also, it results that if $D$ is normal crossing divisor sum of regular divisors $D_i$ ($1 \leq i \leq m$), then the characteristic sheaves are given by:
 $$ C_X= \oplus_{1 \leq i \leq m} \mathbb{N}_{D_i}, C_X^{gp} = \oplus_{1 \leq i \leq m} \mathbb{Z}_{D_i}.$$
All this is in example 3.4 of the manuscript "Géométrie logarithmique" of L. Illusie and A. Ogus.
Could you explain this to me?
Thank you
 A: Here is the basic idea modulo some details. I'm sorry, but somebody else will have to fill in the missing parts about characteristic sheaves.
If we put the trivial log structure on $U,$ then the direct image log structure has sections which are (locally) regular functions on $X$ whose restriction to $U$ is invertible. This follows by definition of $j_*^{log}(\mathcal O_U^\times)=\mathcal M_X$ as the fibre product of the morphisms $\mathcal O_X\to j_*\mathcal O_U$ and $j_*\mathcal M_U\to j_*\mathcal O_U,$ where $\mathcal M_U=\mathcal O_U^\times.$
Now if $D$ is a (reduced) normal crossings divisor, then by definition there exists locally in a neighbourhood of $x$ a (regular) system of parameters $\{t_i\}_{i\in J}$ such that $D=V(\prod_{i\in I} t_i)$ where $I\subseteq J.$ Thus, the local sections of $\mathcal M_X$ are those regular functions $f=ug,$ where $u$ is invertible and $g=\prod_{i\in I}t_i^{n_i}$ vanishes exactly along $D.$
So, for example, if $X=\mathbb A^2_K$ and $D=V(xy),$ then $U=D(xy)=\operatorname{Spec}(k[x,y]_{xy})$ is the plane minus the two axes, and $\mathcal M_U$ is the sheaf associated to $(k[x,y]_{xy})^\times.$ To get $\mathcal M_X$ we consider the (associated sheaf to the) fibre product over $k[x,y]\to k[x,y]_{xy}$ and $(k[x,y]_{xy})^\times\to k[x,y]_{xy},$ which has for example global sections $(a,b)\in k[x,y]\times(k[x,y]_{xy})^\times$ such that $a$ is invertible as an element of $k[x,y]_{xy}$ (equal to $b$), that is, there exists some $c\in k[x,y]$ which we can assume is not divisible by $xy$ such that $ac=(xy)^m$ for some nonnegative power $m$. Then, in say the local ring at any point $x\in D,$ we have $a=u(xy)^m$ with $u=c^{-1}$ a unit, since $c\notin\mathfrak m_x.$ 
In this case, we see that locally the map $\mathbb N_X\to \mathcal M_X,m\to (xy)^m$ essentially determines $\mathcal M_X,$ except that we are missing the invertible $u.$ This is because $\mathbb N_X$ is only a pre-log structure, i.e., the associated log structure $\mathbb N_X^a$ adds the units of the structure sheaf back in for a complete description.
