A log structure on a scheme (or analytic space) $X$ is a (unital) homomorphism

$$\alpha_{X}:\mathcal{M}_{X}\rightarrow\mathcal{O}_{X}$$

of sheaves of (multiplicative and commutative) monoids inducing an isomorphism $\alpha_{X}^{-1}(\mathcal{O}_{X}^{\times})\rightarrow\mathcal{O}_{X}^{\times}$. The monoid structure on $\mathcal{O}_{X}$ is given by multiplication. The triple $(X,\mathcal{M}_{X},\alpha_{X})$ is then called a log space. We often write the whole package as $X^{\dagger}$.