[Nadya]

Where I work, days off are logged but only to make sure you aren't "abusing" the informal policy. Abuse means you get treated more formally and may be told "No, you need to come in today."

Email/call in sick, stay at home. There is an implicit "stay a little later when you're feeling better and/or make up lost time on the weekend". As long as nothing falls behind it isn't an issue. That's the informal policy. If you were ahead or don't fall behind - the extra hours don't need to be picked up. The only important thing is that things remain on schedule.

Hungover, sick, just don't feel like coming in that day, doesn't really matter.