i would go with one app=one database.
it sounds like they have four suites of tables related to a given functionality.
with the current planned design, at some point, you can be sure that some report or functionality will need data combined into a single report.
That is much more easily fixed with your plan for a single unified database.
Otherwise, pulling report data into datatables, merging the data with Linq or something, or else you run into cross database issues, especially with potentially different users for each database.
Based on the name, for the "Logging" database, i bet it's a common log used for all three modules, anyway.
i'd make them use a single database, and just add a naming convention to group suites of tables together
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