How can I trace the exhaust duct of my bathroom fan?
I wouldn't bother with making smoke&mdash;at least not initially.  If this is in an apartment or condo, that might set off a smoke alarm.

The surefire solution is to poke your head up into the attic with a bright hand held light.  If you can't see the bathroom fan area from the access opening, you'll have to crawl through the attic until you are close enough.  Attics are not usually made for careless movement through them:  you might have to temporarily install pieces of plywood, 2x6s 4&ndash;6 feet long, etc. so you can lay a safe trail to get there.  If you don't, you risk falling through the ceiling.

If there is no attic, such as in a multistory apartment building, maybe you could rent an inspection scope (like this) and poke it through the fan into the duct and see what you can see.  There are only two or three possibilities:  1) the duct is clogged or collapsed, 2) the vent box knockout was never removed and not ducted anywhere, or 3) the far end of the duct is pinched or blocked.