For Windows OSs, a quick and dirty trick to log torrent activity and append to a file, would be to use "Run this program when a torrent finishes" or "Run this program when a torrent changes state" functionality (Options -> Preferences -> Advanced -> Run Program), with the following command (modify parameters for echo to suit your needs):
cmd.exe /c /u "echo name(%F)__title(%N)__hash(%I)__state(%S)__status(% M) >> c:\utorrent_activity.log"
Unfortunately, Windows' environment variables (such as: %APPDATA%) won't work here. Also, this will keep showing a command interpreter window for a split second, every time torrents in uTorrent will be changing states, which might be annoying.
Also, one can modify shortcut that starts uTorrent - open shortcut properties, in the "Target" field, add a /logfile switch along with the filename of the file to which application will be logging (see below). Unfortunately, I'm not aware of the option to append to the existing log - it will be overwritten with every application start.
"C:\Program Files\uTorrent\uTorrent.exe" /logfile "%APPDATA%\utorrent\utorrent.log"
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