Remarks on logarithmic \'etale sheafification

We prove criteria for a presheaf on logarithmic schemes to be a sheaf in the full logarithmic \'etale topology and describe several situations where the structure sheaf and logarithmic structure are logarithmic \'etale sheaves. We deduce that the logarithmic Picard group is a stack in the full logarithmic \'etale topology on logarithmic schemes whose structure sheaves satisfy logarithmic \'etale descent.