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We give moduli interpretations using log geometry of the main components of the moduli spaces of polarized spherical varieties defined by Alexeev and Brion. The key new insight is that logarithmic structures alone do not suffice to give a good moduli interpretation. Instead we introduce a notion of an enriched log structure which is a logarithmic structure together with some extra data related to the group action on the varieties being parametrized.

Moduli of spherical varieties and log geometry