They do check your boat registration and the hours better match, as for the log book, any note book will work for you they dont check your log, just what you put on the sea time documentation form which asks for the days per month/ average hrs. per day/ and how many days are inland (which is inside the demarcation line) and how many are offshore (beyond demarcation line). I would be concerned that if you tried to fudge these numbers you could get burned! These guys are pretty sharp and if you put 25 days in June of 1999 and there were 10 days of small craft warnings it won't look so good.
Best advice is to keep and use a log and you won't have anything to worry about!