Remarks on logarithmic étale sheafification

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