Yes, the torn logbook is apart of the Dawnport Quest. Please be sure to talk to NPC Mr Morris to obtain the task.
"The Stolen Log Book
Keyword: log book
Reward: 50 Gold Coins
Notes: You must find a Torn Log Book for Mr Morris. Leave the Outpost either south (Druid) or east (Paladin). Paladins should go south of the ramp to find a cave, druids should go west to find it. Go down to the final floor and fight your way to the north-west end of the cave. You may encounter a Troll Marauder at the end. Do not go up the ladders; instead, walk east across what seems to be a bridge. Open the chest to find the Torn Log Book. Go back to Mr Morris and report your mission.
You can do this task alongside the kill trolls task, which you receive automatically when you kill a troll or by talking to Mr Morris."
Source-
https://tibia.fandom.com/wiki/Dawnport_Quest