  We construct the logarithmic and tropical Picard groups of a family of
logarithmic curves and realize the latter as the quotient of the former by the
algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of
logarithmic abelian varieties over the moduli space of Deligne-Mumford stable
curves, but does not possess an underlying algebraic stack. However, the
logarithmic Picard group does have logarithmic modifications that are
representable by logarithmic schemes, all of which are obtained by pullback
from subdivisions of the tropical Picard group.
