A log is a journal. To "keep problem logs" can mean that the lumberjacks have to keep track of problems they've encountered while logging in a journal. Logs can also refer to a part of a tree that has been cut off. So, "to keep problem logs" can also mean that the lumberjacks have to physically keep the logs that gave them problems. This joke is a play on the double meaning of "keep problem logs".
Lumberjacks have to keep problem logs .