routers can keep logs and so can your modem.
IMO, this day and age, when it comes to public wifi, it is best for the subscriber to keep logs. too many people use public wifi for illegal activity. also, if you share your wifi with a neighbor, you better keep logs. a few years ago in my town, a former police officer shared his wifi with a neighbor. one morning, his door came crashing open and a squad of police officers from the city, state, and fbi, enter his home. search his computer and his home for child porn. fortunately the logs saved his reputation and they traced the mac address to the neighbor.
the reality is, if you use wifi whether at home or in public, logs are being kept.