I just tested, I didn't give moderator permission but rather just gave the test user only the ability to soft delete, he deleted one post and it doesn't turn out in the moderator log. Should he be in moderator group in order I c an see his moderator activity?
A user has to be explicitly made a moderator for actions to appear in the moderator log. Adding to the moderator group doesn't make them a moderator. They have to be added to ACP > Users > Moderators to be a moderator.