Logarithmic de Rham, infinitesimal and Betti cohomologies

Given a log scheme $Y$ over the complex numbers, Kato and Nakayama were able to associate a topological space $Y^{an}_{log}$. We will use the log infinitesimal site $Y^{log}_{inf}$ and its structural sheaf $\mathcal O_{Y^{log}_{inf}}$; we will prove an isomorphism $H^{^.}(Y^{log}_{inf}, \mathcal O_{Y^{log}_{inf}}) \cong H^{^.}(Y^{an}_{log}, \mathbb C)$. The isomorphism will be obtained using log De Rham cohomological spaces $H^{^.}_{DR,log}(Y/\mathbb C)$ along the lines of Shiho. These results generalize the (ideally) log smooth case of Nakayama