Is it unethical for an employee to keep an authorized job log on
another employee when they believe that employee is engaged in
inproper activity?

All of society suffers if large numbers of people cheat, lie and
steal, so I think it is reasonable to make a secret job log of a
co-worker at the request of a supervisor. The supervisor needs to
remember that the secret job log could be a lie. Generally the secret
leaks out and is bad for morale of all who hear about the the stealth.
  Neil

eskimodog-ga:

It depends on what country you are in, and the legal and ethical
situation there.  It is a question for the company's Human Resources
Department and the Ombudsman. Once the ombudsman (or HR Department)
and Executive Management approves, it becomes company policy.  
Eventually, it may be decided by the legal system (if someone files a
lawsuit), but generally companies stand behind to legally defend any
actions taken due to company policy.  Note the courts may find for or
against the employee depending on the situation and the
regulations/law, in addition to company policy.