The main reason Rand and Taim got off to such a bad start was that Taim was thinking partnership, and Rand was thinking "I AM THE DRAGON REBORN!!!! BOW DOWN!" Taim resents Rand for sticking him with teaching the Asha'man, although that is becoming more of an important task. Logain, however. He has no delusions of grandeur. He said to Nynaeve, or Egwene or someone that "I would go to al'Thors amnesty." Having already lost the Power once, he probably just wants to be protected from losing it again. He will take whatever is offered to him, which will probably be a position like a general, or something, because of a) Min's viewing of glory and b) Rand's probable like of him. However, I just can't see him and Taim killing each other. First of all, Taim is a bit stronger in the Power, probably a step above, and he would win if the fought. Where would Logain's glory be then?