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All rational thinkers would clearly know that in a series of arguments anything that appeals to faith would be considered a fallacy.
Yet the rational process itself (the belief that arguments based on sound logic would lead to the correct answer) is itself a belief based on acceptance based on assumption or because it has happened till now.
So does one question the rational process?
The question reminds of the problem with Cartesian doubt and of the Clifford-James debates (somewhat poorly named because Descartes is not actually subject to this type of doubt).
Early in the Meditations, Descartes looks at the consequences for a species of radical skepticism. Here, he looks at the most radical denial of our ability to trust our faculties of reason suggesting that we cannot even trust our ability to reason about our faculties of reason. The problem is that if we reach such a state of non-trust, we also cannot trust our doubts. How does Descartes escape? Well, you should really read the Meditations but briefly he suggests that there's a God who is not an evil demon and isn't manipulating our thought process to the point where we cannot rely on it, and that we have in our thoughts "clear and distinct" ideas which have a special difference from all of our other ideas.
The Clifford-James debate revolves around a claim by W.K. Clifford that we should only believe what we have sufficient evidence to believe. But the problem is that this turns out to be a criterion for which the rule (or whatever level of rule prime) will not be able to pass muster. Williams James is deeply critical of this and suggests that we are all using "faith". But here we need to remember James is a type of pragmatist.
Most rational accounts are going to end up pulling something similar to Descartes -- we are going to need some minimal set of faculties that just plain work to get to any truth. Hume as a type of empiricist and skeptic thoroughly agrees but denies we have such faculties. Kant, on the other hand, thinks that these faculties are implicit in knowledge claims and not problematic insofar as what we say about objects reflects not on the things-themselves but the things as rendered to us through our faculties of understanding and perception (which impose 12 categories and forms of sensibility [space and time] respectively).
Now I take i what Hunan is arguing with you about in the comments is that "belief" and "faith" can have a very wide array of meanings, and it's not clear if you mean what I've answered above or if you mean something else that equates this with what we might call "religious belief" or fideism. There things will get a lot dicier and you're going to have to make a much more robust argument if you want to claim that believing we can trust our sense and our faculties is the same sort of thing as believing in God.
Rationality does not consist in the idea that our ideas should be "based on sound logic". Every argument makes assumptions, including assumptions about the right rules of inference. Those assumptions are not proven because in order to prove them you would need a further argument that would make more assumptions that have to be proven and so on, leading to an infinite regress. Rather, we create knowledge be noticing problems, making guesses about solutions to those problems, criticising the proposals until only one is left and then looking for problems with our new ideas. Rationality consists in being willing to discard any idea that does not survive criticism. See "Realism and the Aim of Science" by Karl Popper, Chapter I.
Some philosophers have tried to criticise this idea by saying things like "you shouldn't throw out an idea as soon as there is a criticism of it." But if you see a criticism and you correctly think you know how to answer it, then when you have answered it your original idea has an addition that takes account of the criticism.
Other philosophers have stated that unless you have justified an idea you can't rely on it to solve practical problems. Since these people are advocating using an idea has not survived criticism they are criticising the standard of rationality described above. The way you should decide on a practical problem is to make suggestions for how to solve it and then criticise them. If your idea clashes with some other idea that has survived criticism that is a criticism of your proposal. If an idea that people wrongly think of as justified criticism then you should take account of it when making practical decisions when it is relevant, otherwise you should not take account of it.
You can criticise rationality, just not successfully.
A very nice articulation of basically this problem is from C.S. Lewis in What the Tortoise said to Achilles.
As a short summary, the Tortoise presents the following to Achilles:
A: "Things that are equal to the same are equal to each other"
B: "The two sides of this triangle are things that are equal to the same"
Therefore Z: "The two sides of this triangle are equal to each other"
This seems like an obvious syllogism, and Achilles agrees, but the Tortoise points out that one could argue for the necessity of accepting:
C: "If A and B are true, Z must be true."
Which is essentially the base of the logic being used. But even if one accepts this, one must then accept:
D: "If A, B, and C are true, Z must be true."
And so on. The point is that one can continue this forever, and even with all these assumed premises one needn't accept the conclusion. Ultimately, as you say in your question, it comes down to a matter of giving up on all the technical premises and just saying "Z must be true because of the above" regardless of the hidden assumptions beneath that.
