An axiom is simply a primitive sentence of a language system. It's often used in two different contexts:
Context 1. Sentence s is an axiom if and only if ∅ ⊢ s.
Context 2. Sentence s is an axiom if and only if ∅ |= s.
The first context is that of syntactical language systems (i.e. proof systems). There, the main logical relation is that of syntactic consequence ( ⊢ ). A sentence in such systems is said to be an axiom just in case it's a syntactic consequence of the null set of premises.
The second context is that of semantical language systems. There, the main logical relation is that of semantic consequence ( |= ). A sentence in such systems is said to be an axiom just in case it is a semantic consequence of (or is logically entailed by) the null set of premises.
As David H already mentioned: in both of these contexts, an axiom is a sentence that does follow from a set of other sentences (namely: ∅) by a number of logical steps (namely: 0). This, of course, trivially.
Lastly, in the context of philosophy of mathematics, a certain set of axioms might be called "logical" simply to distinguish them from axioms governing the use of what we ordinarily take to be 'extra-logical' notions. For example, a set of axioms for propositional logic may be designated as "logical axioms" to distinguish them from a set of axioms for arithmetic or set theory; these other sets of axioms in this context may then be designated respectively as "arithmetical axioms" and "set theoretic axioms."
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