How do you argue with those that do not go by logic?

Try to elicit a sympathetic emotional response.

Usually they will try to insult you first.  And that's their first fallacy (an ad hominem), you can then take that opportunity to make them look stupid, usually by calling them out on it, and perhaps they will back off... but if they get way too emotional, you're better off saving your time worrying about more important things...

You don't. Waste of time. 

You don't argue with them, for such a thing is futile. Arguing with someone who does not utilize logic is illogical.

I do not argue with such people.  I avoid such people.

I don't.  I allow the person to say their piece three times say mine Once (rarely twice) and leave.