Logic sprouted of different branches and now there is for computers, but as it grows the symbols tend to deal with problems not in accordance with what central philosophy tackles. It should therefore be specific to philosophy if the study is philosophical reasoning which is philosophical logic. Philosophical logic is asserted (by academic references) as the application of formal logic to philsophical reasoning, thus if so, is it formal logic that is the most important element of logic in terms of philosophy?
There is no single agreed upon axiom or theory of logic. Even the traditional laws of thought, such as the principle of identity and noncontradiction, are disputed by logics based on dialetheism (i.e., paraconsistent logics). There are of course different logical theories that are more or less appropriate given the context and subject matter. The particular axioms of Zermelo–Fraenkel set theory, for example, are widely agreed upon as the most common foundation of mathematics. However, we need some way of determining whether the axioms and their associated theoretical assumptions are correct. For this goal we would appear to need some independent, or higher standpoint from which we can judge the acceptability of different proposed logics.
Contrary to the prevailing orthodoxy in the modern West, not everyone subscribes equally to the mathematical turn to symbolic logic. Historically speaking at least, it is dialectical logic that has held center stage in philosophical reasoning; a type of logical theory that has been largely rejected since the rise of positivism and analytic philosophy. However, in philosophical terms, dialectic plays a higher, more synthetic role than purely analytic or mathematical formalism, which is generally built up from calculi expressing fixed identity statements that are derived inductively or a priori. In other words, these logics represent a type of epistemic and semantic atomism. Dialectical logics, by contrast, are contextual and organic and deal with the movement of an objects identity within a total theory or system of thought. These logics thus represent a type of epistemic and semantic holism.
The former trend was present in embryonic form since Aristotle, the Medieval logicians, and more significantly in the Kantian critical philosophy. It reached maturity in the modern turn toward symbolic and propositional logic, which can be easily observed in the works of scientifically or mathematically inclined philosophers, such as Richard Avenarius, Gottlob Frege, Bertrand Russell, the early Wittgenstein and Kripke, and so on. The latter movement, however, was also present, and some would say to a larger extent, throughout the history of philosophy. The modern rejection of this type of logic has its roots in Kant's critique of dialectical illusion, and the supposed excesses of Hegelian philosophy. It was this German Idealist strand in 19-20th century British philosophy that the analytic school reacted against. However, there has been something of a turn, or at least, there is now a possibility of domesticating Hegel for analytic consumption. This movement began with the early Pragmatists, such as Charles Sanders Peirce, James Dewey, the late Wittgenstein, and Wilfrid Sellars, and is now a far more commonplace position, as can be seen in the works of post-analytic and neo-pragmatist philosophers such as C. I. Lewis, W. V. O. Quine, Donald Davidson, John McDowell, and Richard Brandom. As mentioned earlier, of course, there are also paraconsistent logics to consider, which can be seen as an attempt to formalize traditional dialectics and defend the productive nature of logical paradoxes.
The reason I bring up dialectics and holistic logical theories in response to the original question of whether formal logic is "the 'most essential element' to philosophical logic," is because I believe the nature of philosophy is incomplete without its speculative and systematic counterpart to the subject matter. A purely formal and rigidly analytic, or mathematical type of logic is, in philosophical terms, a lower and more artificial level of rationality. It is the least interesting and revealing kind of logic since it lacks the dialectical plasticity and systematic movement that characterizes, but also justifies, the concepts of philosophy and the nature of true being. For this reason, dialectics should at least be accorded some measure of respect as an integral component of philosophical thinking. Regardless of the current popularity of logical formalisms in mathematics, the empirical sciences, and computation, I think it's far from settled that formal logics are the best way to conceive of rationality, or even a sufficient account of philosophical logic.
To supplement the already given answer in a similar direction (although the post is a bit old), which touches many important issues, is the fact that each domain might (and usualy does) require its own logic (in the sense of descriptions of its own processes and interactions, whether formalised or not is another matter) and usualy these domain-specific or organic logics are not of the classical flavor.
This is manifested in modern mathematics and physics as well (surprisingly) in domains or toposes (places, spaces) which have their internal/organic logic of the processes and interactions and these are different from classical logic (mostly of the intuitionistic flavor, where the law of excluded-middle is actually rejected). However this should not seem surprising, that a positive science like physics or mathematics arrives at such notions, since if there is even a remote chance of this being realizable then all forms of knowledge should encounter it at some time.