According to Aristote, Prior An., 24a10-24a15: "First we must state the subject of the enquiry and what it is about: the subject is demonstration (ἀπόδειξιν), and it is about demonstrative understanding. Next we must determine what a proposition is, what a term is, and what a deduction (συλλογισμός)."
Here we have already the basic "objects" to be enquired by logic: proof, deduction, language.
"Traditionally, (formal) logic is concerned with the analysis of sentences or of propositions and of proof with attention to the form in abstraction from the matter" (Alonzo Church, 1956).
Logic is formal: "The discovery of the formal treatment of logic, i.e. of the possibility of describing deductive reasoning with sentences in terms of their form, appears with Aristotle" (S.C. Kleene, 1952).
Modern logic is mathematical: "The subject of formal logic when treated by the method of setting up a formalized language is called symbolic logic, or mathematical logic" (Alonzo Church, 1956).
And see Jean-Yves Beziau, Why Logics (2023): "The substantive 'logic' is used in three different ways:(1) reasoning;(2) a system describing reasoning, that is, 'a logic';(3) the science studying reasoning through the development of logical systems and tools to study them."