There is indeed nothing in that log. You could also open up the log manually in game and copy-paste that. The hotkey is F9 - if it doesn't work then open up the hotkeys window with F1 and search for 'log'. Make sure you do this on FAF as the hotkeys window doesn't exist in Steam.
Play a game on FAF for a few minutes then copy the content of the log. Works best when you're playing windowed mode.