Although logos is far more prevalent throughout The Republic, pathos is used intermittently to both promote and indict various arguments. In fact, part of the reason Socrates argues that art and poetry need to be outlawed in the ideal state is because of their ability to evoke strong emotions in the audience, which may in turn inspire certain conduct: in other words, their potential for pathos.
In Book 10, Socrates describes the pathos of Homer quite succinctly:
[...] my earliest youth had an awe and love of Homer, which even now makes the words falter on my lips, for he is the great captain and teacher of the whole of that charming tragic company; but a man is not to be reverenced more than the truth.
Homer's tragic poetry inspires a love for Homer within Socrates; Socrates feels the push and pull of Homer's characters and the trials and tribulations that constitute their story. However, "a man is not to be reverenced more than the truth," or in other words, true wisdom and knowledge are to be revered more than anything else. It is precisely this effect of poetry—its ability to move the emotions—that is why it must be outlawed in the ideal state according to Socrates (though, notably, Socrates's very argument here—the "words falter[ing] on [his] lips"—can be read as an instance of pathos). The alleged pathos of Homer highlights just how dangerous art can be and why it must be heavily restricted in the ideal city.