Example
Let \(A = k[P]\) with \(P \to A\) a chart for a monoid \(P\), \(p\mapsto \chi^p\), over \(\Spec k\) with trivial log structure. We study the log differentials \((\theta, \delta)\)to \(A\) and the log tangent sheaf of \((\Spec A, P)\).
Differentials: Choose \(\delta:P^{gp}\to A\) to be an arbitrary morphism. Then \(\theta(\chi^p) := \chi^p\delta(p)\) satisfies the Leibniz rules because \(\delta\) is additive, and extending determines \(\theta\). So choosing \(\delta\) forces \(\theta\). This means
\begin{align*} \Omega^1_{\Spec A/\Spec k}(\log) = A\otimes_{\mathbb Z} P^{gp}, \quad T^{\log} = A\otimes_{\mathbb Z} N \end{align*}where \(N = \Hom(P^{gp}, \mathbb Z)\) and \(\partial_n(\chi^p) = \langle p, n\rangle \chi^p\).