I personally would keep them seperate.
FAA instructors in my limited experence have a nasty habit of writing all over the bloody thing. It is apparently there right!!
And apart from that if your main log book goes missing in tranist your a bit stumped.
I would transfer a page into a cheapo log book then use that to get your FAA stuff put in then transfer the hours back into your main log book when you get home. Saves having scribbles in your pro log book for interviews.