The Log is where everything you do, all your activity in the game, is stored and displayed as an editable text file. Logs are a Hacker's best friend if they know how to use them and also their worst enemy if they don't. Removing your activity from other player's logs is vital to a successful hack and reduces the chances of that player finding you and getting their revenge.
Anyone can edit any Log. That's both a fun and scary thought. When editing your Log or another player's Log you can do anything you want. You can manipulate it, remove activity from it, give a personal message, or leave your Hacker calling card (if you're the type and good with ASCII).