Your Log Book contains a record of many things.
Primarily, it records the status of all quests in a game. At any given moment, the list of quests represents the then current status of all quests – regardless of whether they were started or completed by your character or by another character, and also regardless of whether the actions took place while your character is/was in the game or not yet arrived. Hint: If your character is a Trader, check to see if Goods Messenger is done when you enter the game. If so, Bravemart will be open. The quest log is reset for every game.
The Log Book also records all Scrolls that you read as you progress. This record is not reset, but remains cumulative for your character.
There are other tabs of interest as well.