My basement shower exhaust fan was vented into the area above the suspended ceiling. I want to vent it outdoors, the way it should have been done the first time. The shortest way out is through the rim joist about 2 feet away. I can't go through the one floor joist in my way to get there. Is the better solution to drop the ceiling in the shower area enough to allow the fan exhaust to go under the floor joist and out the rim joist?

Depending upon the size of the floor joist and the local building codes sometimes you can cut a perfect hole the size of the pipe right through the joist. Especially if these atr tru-joist or manufactured joist. As mentioned before contact your local building department with your dimensions.

New at this, not sure if this is the way to follow-up. Thanks for your response. I think the exhaust requires a 4" hole, my joist are 2x10, didn't think I could make the hole. That is why I was considering going under.

Sorry I did not provide the answer you wanted to hear. Go with the option you wanted to start with and drop the ceiling. It is your labor not mine. Good luck.