... in the end they always can be rejected by use of reason and logic ...
Logic and reason can't explain Love, compassion, empathy, or natural disasters, life, death, or road rage, drunkenness, drugs, murder, rape, etc. Any act committed though passion can't be explained by logic and reason. The mind is very conning and conniving, it can rationalize any atrocity against his fellow man, from throwing millions into gas chambers or blowing themselves up as martyrs.
No, the atheist who co-ops Logic and Rationalism tries to make them solely their own, as if the rest of Humanity is lacking this mental acumen. Logically the atheist can rationalize the need to exterminate all religious believers; he can reason that he is saving Humanity, whilst conveniently forgetting that some madman can push a button and destroy all life on Earth.
An appeal to logic and rationalism always fails when faced with violence. Then he is most likely to drop the facade, say "The hell with logic and reason," and lash out at his attackers, reverting to his basic animal nature, the survivalist instinct. Logic and reason are what elevates man beyond his animal nature, but they are not inherent qualities which govern his whole personna, they are merely facades because humans are sensate animals, and when push comes to shove, the animal will over rule logic and reason. Logic and Reason are merely appeals, hardly different that the religionist appealing to a God, as he did from the very dawn of Humanity. Logic and Reason are social constructs, hardly any different from the 10 Commandments, upon which Western Society built their Laws.
So,
start with the 10 Commandments. Dismantle them with Logic and Reason. I'll even allow you to start from conclusions - what the world is like today. Show me how doing away with each and every Commandment (made then) would have made a different and better world today. Prove to me how never having had the 10 Commandments would have directly led to the Revolution of 1776 and the world as it is today, staring with the world as it was some 10,000 years ago. Logic and Reason will not allow you to manufacture "what-if" scenarios, as that would be imagination run amok.
Chances are that all you will be doing is making a case for license, the license to do anything we want regardless of whether or not it hurts another - and if truth be told it must hurt another. So through the logical advocation for License you will ultimately make the case for animal instinct.
I doubt that you can start the argument by saying that since animals don't have religious rituals (assumptive conclusion on your part) that we as Humans don't need it. The fact is that some animals mourn their dead, which is a prelude to religious ritual, like the ritual for the dead. The fact is that most mammals "seem to" have consciousness, exemplified by their knowing when they are about to die. We've all seen elephants giving respect to their dead, no? Cats have been known to go be with their owners right before they die, for example.
Using logic and reason, without co-opting the argument (for example, saying, "Everyone knows that one should not kill, so that's not religious" - which begs the question, "If everyone knows not to kill, then why did it need to be written down?") refute the 10 Commandments.