How does Socrates use ethos, pathos, and logos in his argument?
Socrates's arguments in Crito are framed as a direct response to Crito's pleading with his old friend to make good his escape. Crito has such great love and regard for Socrates that he cannot stand to see him put to death, especially on such unjust grounds. But Socrates is having none of it; he will stay and take his punishment. As it may seem strange for him to do so, he proceeds to explain his reasoning.
Ethos: This is a rhetorical device which is designed to appeal to an audience's moral values. Not surprisingly, Socrates makes fulsome use of moral appeals in Crito as his whole philosophy was devoted to the question of what constitutes the good life. And here we see Socrates emphasizing the qualitative nature of life. It is no use his friends being upset over his imminent demise, what matters is whether one has led a good life. As a good life involves not knowingly doing harm, then there's no sense in which such a life can ever be truly undone by an unjust punishment. A good life transcends whatever anyone can throw at it. And Socrates, by his own high standards, has indeed led a good life.
Logos: An appeal to one's reason. It's vitally important for Socrates, a philosopher, to frame rational arguments in response to Crito. Socrates chides his friend for using arguments shared by the multitude, by women and children. Difficult though it may be for his friends to accept Socrates' decision, he does have rational arguments to justify his staying to face the music.
Socrates is an Athenian citizen. He has lived in the city all his life. By doing so, he has tacitly accepted its laws. If he were to run away now, then he would be breaking an unwritten agreement between himself and the state. What right has Socrates, or anyone else for that matter, to defy the law? If Socrates had wanted to leave Athens he could easily have done so, but chose not to. In choosing to remain, he made an undertaking to abide by Athens' laws, for good or ill.
It would be completely contrary to everything that Socrates has ever stood for, for him to use pathos, an appeal to one's emotions. Socrates is a thoroughgoing rationalist, believing that emotion has no place in the examined life of the philosopher. It is the preserve of the ignorant and ill-educated. Any emotional content we find in Crito is supplied largely by Socrates' eponymous friend.