Definition
Let \(X\) be a scheme. A pre-logarithmic structure on \(X\) is a sheaf of monoids \(\mathcal M_X\) together with a morphism \(\alpha:\mathcal M_X\to \mathcal O_X\) of sheaves of monoids, considering \(\mathcal O_X\) a monoid under multiplication.
A pre-log structure \(\alpha:\mathcal M\to \mathcal O_X\) is called a logarithmic structure on \(X\) if \(\mathcal M\) contains a copy of \(\mathcal O_X^\times\) on which the restriction of \(\alpha\) is an isomorphism; i.e. if \[\alpha|_{\alpha^{-1}(\mathcal O_X^\times)}:\alpha^{-1}(\mathcal O_X^\times)\to \mathcal O_X^\times\] is an isomorphism.
A log scheme is a scheme with a log structure.