Log continuity of solutions of complex Monge-Amp\`ere equations

Let $X$ be a compact K\"ahler manifold whose anticanonical cohomology class is semipositive. Let $L$ be a big and semi-ample line bundle on $X$ and $\alpha$ be the Chern class of $L$. We give a sufficient condition ensuring that the solution of the complex Monge-Amp\`ere equations in $\alpha$ with $L^p$ right-hand side ($p>1$) is $\log^M$-continuous for every constant $M>0$. As an application, we show that every singular Ricci-flat metric in a semi-ample integral class in a projective Calabi-Yau surface $X$ is globally $\log^M$-continuous with respect to a smooth metric on $X$.