I think Islam is based on logic and then faith plays a great role later on, but the actual base of the religion is logic and reason.
For example, the Islamic Tawheed conception of God is clearly the most logical and most monotheistic of any belief system. Since it rejects illogical concepts such as divine incarnations, the holy trinity, anthropomorphism, etc., it shows that it is based in logic instead of previous religions such as Hinduism and Christianity which have traces or are made up greatly by older traditions with some other new beliefs mixed in. Therefore the Islamic concept of God is the most logical of any faith (same for Judaism since their concept is basically identical).
Also, many ancient peoples performed illogical and abominable practices such as idol-worship, astrology, fortune-telling, palmistry, card-reading, etc. But when the light of truth shines upon these practices, it is apparent that they are all incorrect and superstitious - And of all the religions I have studied, Islam does the best job of discrediting and condemning such practices which is another major reason why Islam is based on logic.
Also, the method of dawa employed by such great da'ees as Zakir Naik, Ahmed Deedat, Yusuf Estes, etc. are logic-based. By that I mean that those da'ees point to the logic of the Islamic concept of Tawheed, the scientific facts within the Qur'an, the health reasons behind why haraam foods are avoided, etc. On the other hand, Christian evangelists try to lure people to Christianity by talking about Christ(PBUH)'s love and being saved from hell by the grace of God, and other such emotional tactics instead of trying to appeal to logic and reason like most Islamic da'ees I have heard do.