Logos is one of the most overdetermined words in Greek philosophy, and it carries different content in Heraclitus, in Plato, in Aristotle, in the Stoics, and in John's gospel. For the Stoics specifically, logos names two things at once: (1) the rational structural principle that pervades and orders the entire cosmos, and (2) the rational faculty in each individual human being, which is a fragment of that cosmic principle.
This double identification is doing important work. Because the logos in you is continuous with the logos of the cosmos, the cultivation of your own rationality is not a private project — it is the work of aligning your fragment with the whole. To live according to nature, in the Stoic phrase, is to live according to logos — yours and the cosmos's, which are continuous.
The Stoic logos is not personal in the way the Christian Logos would later become (the divine Word made flesh). But it is also not impersonal in the modern scientific sense. It is providential — the cosmos governed by logos is a rational order tending toward the good, even where individual events appear locally bad. This providential ordering is also what underwrites sumpatheia, the mutual sympathy of cosmic parts. Cleanthes addresses logos as Zeus and addresses Zeus as personal. Later Stoics — Seneca, Marcus Aurelius — sometimes hold the language loose, addressing Nature or Providence or the gods more or less interchangeably with logos.
The practical implication. When the Stoic says live according to nature, the nature in question is rational nature — both yours and the cosmos's. The instruction is not to live a primitive or unreflective life. It is to live the life appropriate to a being whose distinguishing feature is participation in logos — which is, for the Stoics, also the human telos. The practice of philosophy is, on this picture, the cultivation of one's continuity with the rational order — and the misalignments of practice are partial separations from it.