you can't drive the car on a public road without tax, unless you are taking it to a pre arranged MoT or you are driving it on company trade plates.
as for the log book, i personally wouldn't buy a car without one, because you will have no service history. who knows when it was last serviced, or if it's ever been manufacturely serviced ever? you could drive the car 2 miles down the road and it may seize or something.
i wouldn't buy a car with no log book. you can if you want, but i wouldn't expect it to last very long. as someone said, who knows how it's been looked after, and it could be stolen.