Yes, quite so. By way of clarification, logic is a process, not a result, and so one may be perfectly logical, even in proceeding from false premises. One can even validly arrive at a true conclusion with false premises, but it's not guaranteed, and it's unsound logic even if you do.
And there's the rub - how do you know your premises are true, that your underlying axioms are true? Well, you rolls the dice, and you takes your chances, pretty much ;)
Two ways:
1. If the premises you use lead you to conclusions that are contradictory, if your reason (logic) is otherwise correct, one or more of your premises is incorrect.
2. Your ultimate premises must ultimately be axioms, those irreduceable primaries of existence and consciousness. Axioms are not assumptions, they are, like all knolwedge, discoveries about the nature of existence or our knowledge of it, that once discovered, cannot be denied without being self-contradictory. For example the absurd post-modernist assertion that true knowledge is not possible. The axiom this contradicts is, "knowledge is possible." If knowledge were not possible, nothing could be known, including the assertion that knowledge is not possible. The expression, "true knowledge," is redundant.
Hank