Only the "Quest Log", not the "Duty Log".
Basically, if you're in dungeon, you'll only see the duty for the dungeon, no more quests.
On my PS4, if I'm in a long dungeons and have a whole bunch of long quests, my log takes the whole right part of my screen, and overlaps with my crossbar and my stats.
So I did a macro to toggle duty info on/off.