Venting of bath fans should be done through a gable end only and if a gable end is not available, the fan and vent installation should be done thru a soffit framed and drywalled below the 8 foot ceiling level.
This way the vent hood termination can be thru a side wall....and similar to venting thru a gable end.
Venting thru an eave soffit or thru the roof is only asking for trouble....with moist air pulled back up into the attic via open soffit vents below eaves...and with condensed moisture running down the duct and into the fan with roof mounting.
Most codes require at least a 50CFM (cubit foot per minute) air exchange in a bath with no windows and a 30CFM in a bath with a window.