If you've just finished your level 15 quest, then I assume this is for something other than DRK (and I'm guessing that you boosted DRK).

The hunting log only needs to be unlocked once, after which it is immediately available to all other classes, so it's possible that boosting a job unlocks the log automatically.

As Enkidoh said, have a look in your Logs menu to see if the hunting log is already unlocked.

If it's working, you'll see hunt targets marked with a yellow symbol over their name - but only when you're equipped as the class that needs to hunt them.