From the looks of it, you seem to have boosted AST to 70 and, I'm assuming, abandoned whatever class you originally started as. The hunting log is normally unlocked after your level 10 class quest, which will also unlock the ability to take on other classes/jobs. You've bypassed the job limitations, but the log unlock requirement is still tied to, presumably, the level 10 class quest for the class you initially started the game with.

Just a hunch. Not 100%. Pretty sure this mostly only comes up when players boost an expansion job which doesn't have any hunting log, and therefore no reason or means to unlock it. Also going to assume that probably players that boost a base ARR job don't run into this issue, since the quest necessary for unlocking is flagged as complete.