You may actually be hourly and not salaried. That said, it's not uncommon to be forced to log when you work, especially when there are PTO and things like that involved and that need to be accounted for.
Are you paid for the hours you work in the internship? If so then you are not an exempt salaried employee.
Are you paid for the hours you work in the internship? If so then you are not an exempt salaried employee.