you use the /usr/bin/script program to record a transcript of a session. no term emulators that i know of provide this kind of feature natively though.
Having script run in your shell when it starts breaks X login sessions. Most terms do this actually, just no on Linux ironically. e.g. iTerm2, Terminal.app, Putty.
GNU Screen can also do logging, if that helps the OP.
Yeah, I use tmux though. I wish I had cycles to add it to tmux.