Today I had a Necromancer in hand, wondering whether to play it, since I had other terminals and 1 action left. But the only way for me to know what's available in the trash, seems to be to scroll through the log and check what I already used. It's pretty awkward, especially since I can't even see the log and the trash at the same time.
Of course the face-down cards in the trash are still open information, so the best would probably be to mark the used cards somehow.