The battle log occupies a fair amount of real estate in the top left corner of my screen, even in full screen mode. On a laptop or a tablet, it's taking away a decent chunk of my battle field.
Is there a way to turn this log off or hide it?
The top left corner of the screen? Maybe it's chat you're talking about? If so, then there is a way to hide it. "H" key switches between three modes (always visible, auto and always hidden) if I'm not mistaken.