There are 3 hunting logs per day. at level 53 (1), level 56 (2), level 3 (59 i think) and there are the weekly hunts unlocked at level 60. Don't count me on these numbers might be off by 1 lol, anyways you can join hunts for A and S ranks at any time, however only at level 60 they rewarding the full amount of law tome stones. If you're not level 60 you still receive full amount of seals and poetics.

You want to save your centurio seals for level 60. That same npc you see the minions at, will sell you 3 items to upgrade your ilvl 170 armor to lvl 180 at level 60!

Hunting logs reset everyday at 8 am pst, if you picked up a stack lets say at 7:30am that are the logs from the day prior. So at 8 am you will be able to pick up another stack.