log book indicates the 'registered keeper'! but unless you have the money to insure it or the inclination to ride it, hand it over and enjoy your life?
Like some of the other folks have said, it's whoever holds the receipt or can prove that they paid. One point to consider though, if it is registered in your name and he uses it, you will be responsible for payment of speed camera fines, taxing the bike or making sure it is sorn. so i would sort it out quick or at least make sure it is registered to him or he could stitch you up with a few speeding tickets, which you will have to prove was not you, rather than him prove it was.