Example
Let \(X\) be smooth over \(k\) and \(D\subset X\) a normal crossings divisor. Endow \(X\) with the divisorial log structure \(\mathcal M_X\), and give \(\Spec k\) the trivial log structure. If \((\theta, \delta)\) is any log differential, then \(\delta\) is determined by \(\theta\) and hence
\begin{align*} T_X(-\log D)(U) = \{\theta \in \sDer_k(\mathcal O_U) ~ \mid ~ \theta(I_D) \subset I_D\} \end{align*}where \(I_D\) is the ideal cutting out \(D\) in \(U\). In coordinates we can write \(D = \{x_1 \cdot ...\cdot x_r = 0\}\), and then writing \(\theta = \sum a_i\partial_{x_i}\), the condition above then says that
\begin{align*} a_i \in (x_i), \quad i\leq r \end{align*}since
\begin{align*} \theta(x_1\cdot...\cdot x_r) = \sum \frac{x_1\cdot ...\cdot x_r}{x_i}a_i \in I_D. \end{align*}To see this, quotient both sides by \(x_i\) to get \(a_ix_1...\widehat x_i...x_r \in k[x_1,...,x_n]/(x_i)\). Since \(\theta(x_1\cdot ...\cdot x_r) \in I_D\subset (x_i)\), we know \(a_ix_1...\widehat x_i...x_r - 0\). But the quotient ring is a domain with \(x_1...\widehat x_i...\widehat x_r\neq 0\), so \(a_i = 0\).
This means the sheaf is locally free generated by
\begin{align*} x_1\partial_1,...,x_r\partial_r,\partial_{r+1},...,\partial_n, \end{align*}dual to the frame
\begin{align*} d\log x_1,...,d\log x_r, dx_{r+1},...,dx_n. \end{align*}