Seeing that you have so many dogs, you would probably release them to kill me if things go out of hand. I would rather leave than getting mangled alive.
But, my dear fellow, you've only just come. You've had no hunting -
Logos
No, I will not hunt with you. What if I die? Won't it be a shame if you killed the person who wrote all those hunting books you've read?
Tonight, we will hunt - you and I.
Pathos
I don't think I would want to. With my experiences in the war, I have come to realize that human life is important. One cannot just throw their's away. So, I don't want to participate.
Ethos
You will find this game worth playing.
Rainsford is using logos in his response because he uses logic to connect the room with dogs to what might happen during the hunt.
Rainsford is using pathos in his response because he is trying to make Zaroff feel guilt and shame in an attempt to have him end the hunt.
Rainsford is using ethos because he uses his experiences and reputation as a veteran in his attempt to convince Zarboff.