Definition
Let \((X,\mathcal M_X)\) be a log scheme over \((S,\mathcal M_S)\). The log tangent sheaf of \(X/S\) is
\begin{align*} T^{log}_{X/S} = \sHom_{\mathcal O_X}(\Omega^1_{X/S}(\log), \mathcal O_X), \end{align*}i.e. it’s the dual of the sheaf of log differentials, as you might expect. It’s also the sheaf of log derivations into \(\mathcal O_X\); the pair \((d, d\log)\) is the universal log derivation, so for every \(\mathcal O_X\)-module \(E\),
\begin{align*} \Hom_{\mathcal O_X}(\Omega^1_{X/S}(\log), E) \xrightarrow{\sim} \sDer^{\log}_{X/S}(E), \quad \phi\mapsto (\phi\circ d, \phi\circ d\log). \end{align*}Taking \(E = \mathcal O_X\) then gives \(T^{\log}_{X/S} = \sDer^{\log}_{X/S} = \sDer^{\log}_{X/S}(\mathcal O_X)\). So “log tangent sheaf” and “sheaf of log derivations into \(\mathcal O_X\)” are the same thing.
Remark
To view \((d, d\log)\) as the universal derivation, we think of \(d\log\) as a map
\begin{align*} d\log:\mathcal M_X^{gp} \to \Omega^1_{X/S}(\log) \end{align*}defined by \(d\log (m) = (0, 1\otimes m)\).