▻ OUTLINE OF EPISTEMOLOGY AND LOGIC
◆ Epistemology (logos tes epistemes) is the philosophical inquiry into knowledge : the origin of knowledge, the place of experience in knowledge, the different kinds of knowledge, the relation of knowledge to certainty, the possibility of knowledge (the challenge of scepticism). These and such like matters.
◆ Logic (logike, logos) is concerned with the rules of valid inference, the forms of valid inference (modus ponens, modus tollens, the syllogism, &c.) and with relations of implication, consistency, contradiction & self-contradiction, and independence that hold between propositions or sentences. Rough and unpolished as this characterisation is, it is broadly accurate. It is also broad enough to accommodate the Questioner's explication of 'logic'.
▻ INTER-RELATIONS - YES
Given these accounts of the two inquiries, there are certainly inter-relations between epistemology and logic :
☛ I can't know that p is true if p is a self-contradiction, for one example. I can't know that S is both p and not-p in the same respects at the same time, for another.
☛ Equally I can know that modus ponens is a valid rule of inference.
☛ There can be and is an epistemology of logic, asking how we know or can know e.g the validity of rules of inference.
☛ And there can be and is an epistemic logic, which represents epistemic relations logically :
Ka (φ → ψ) → (Kaφ → Kaψ)
That is to say that if a knows that φ implies ψ then if a knows that φ, this implies that a also knows ψ.
I'm not entirely sure about this. Suppose a fails to draw out the implication for whatever reason ? But I don't commit myself to the example; I use it just as a case of how epistemic relations can be 'logicised'. It does, however, introduce another interrelation : my 'logicisation' example, when a does draw out the implication, relies on a's knowing the logical consequences (or at least this logical consequence) of what he knows.
▻ IDENTITY - NO
So there are cross-connexions. But I can't see that logic and epistemology are identical - one and the same inquiry.
◆Logic as formal
This is because logic is formal and in certain respects independent of truth. I'd better be very careful in explaining what I mean by this. Take the following example :
Napoleon was a Spaniard
All Spaniards are over six feet tall
Napoleon was over six feet tall
Truth comes in because the conclusion cannot be false if the premises are true. But both premises and conclusion are false. If logic can generate falsity in this perfectly valid way, it hardly facilitates the acquisition of knowledge since knowledge cannot be false.
◆Logic only recycles what is already known
The identification of epistemology also fails insofar as the syllogism does not and cannot generate knowledge :
All men are mortal (suppose we know this)
Socrates is a man (suppose we know this)
Socrates is mortal
The conclusion, which follows validly from the premises, recycles but embodies no knowledge not present or implicit in the premises. The conclusion of a syllogism, if not this one, might be psychologically surprising but it contains no new knowledge.
Simon Blackburn, Oxford Dictionary of Philosophy, Oxford : OUP, 1996, 123 & 221-2.
R. Audi, ed., The Cambridge Dictionary of Philosophy, Cambridge : CUP, 1996, 233-8, 440-5.