I have created two DbContexts, one is for application configuration, the second is for logging.
The reason being, I want to set a maximum size on the logging db so
Yes you can do it. You just need to initialize before you move on to the next one.
Database.SetInitializer(myInitializer);
MyDbContext context = new MyDbContext();
context.Database.Initialize(false);
Database.SetInitializer(myInitializer);
MySecondDbContext context2 = new MySecondDbContext();
context2.Database.Initialize(false);
Note: that I usually get the DbContext instance from a dependency resolver...