I’ve never seen a CFR that states engineers must keep a log of routine operations which doesn’t mean such a CFR doesn’t exist. However, most insurance companies and classification societies do expect machinery log to be kept. Not having a log book would raise suspicion.
A question arose recently regarding the log included within one companies ISM program. They had a daily ‘boat log’ which was on the computer, a copy of which was printed each day and put into a binder. They actually thought this would suffice for an official log.