Short version: ministries do stuff. Logarchies support ministries in the doing of stuff.
Longer version: A ministry, whether one of the big Ministries of the Throne (headed by a Minister of the Throne who sits on the Council of the Star) or one of its smaller subordinate Ministries of the State (headed by a Minister of the State who sits on the Council of Ministers) is an organization which goes about the business of the Empire. They’re the people who actually run the treasury, the Imperial Military Service, the diplomatic corps, the Emergency Management Authority, etc., etc., etc.
(There are also the Shadow Ministries which aren’t part of the formal government at all, but that’s a whole other question.)
A logarchy provides services to ministries. There are lots of civil-service support functions that the ministries, etc., all share, and it would be pointlessly inefficient to duplicate these functions all across the Service. Instead, everyone makes use of central function suppliers such as the:
- Central Office of Records and Archives
- Imperial Services (the people who run public-facing offices)
- Logarchy of Engraving, Printing, and Stationery
- Logarchy of Pay and Rations
- Logarchy of Procedures and Methods
- Logarchy of Procurement, Storage, and Logistics
- Logarchy of Promotions and Appointments
- Logarchy of Protocol, Ritual, and Symbology
- Logarchy of Public Architecture
- Logarchy of Statistics and Computation
- Loyal Association of Pages
…who provide the associated functions to everyone who needs them.