To get the log in Gremlin Village, Mickey must hit the valve on Small Pete's ship, get the log, and show it to Bennet. If Mickey gives it to Shaky, he will give the mouse the Gremlin Pin, but then the log will be gone and the Quest cannot be completed.
When Mickey meets Small Pete in the Colosseum, if he showed the log to Bennet, Small Pete will give the mouse the Inky Mickey Extra Content and a Bronze Pin. If Mickey gave the log to Shaky or ignored it completely, Small Pete will attack him with a Blotling Horde. When Mickey goes to Mean Street, Big Bad Pete will give him the Small Pete Pin and some E-Tickets if the log was showed to Bennet. If it was traded or ignored, Pete will give him nothing and be angry about it.