Unlock Hunting Log

[Enkidoh]

What class specifically? Each class gets their own Hunting Log upon completing the first class quest for each upon joining each Guild. Classless Jobs like DRK do not have Hunting Logs however (the actual Hunt, unlocked from the notice board in each GC HQ, outside the tavern in Ishgard from level 50, and on a wall in Rhalgr's Reach and outside the teahouse in Kugane from level 60ish, replaced it).

Are you looking in the right place? The Hunting Log can be found under the Logs menu and then clicking on the relevant class logo (when you join a Grand Company you're also given a seperate Grand Company Hunting Log which rewards GC seals instead of EXP and which can be selected from the same screen by clicking the nation flag of the GC).