This shows Ethos because it appeals to the speaker's status and authority. With MLK saying he is a minister like the clergymen he wrote the letter to.
I say this as a minister of the gospel, who loves the church; who was nurtured in its bosom; who has been sustained by its spiritual blessings and who will remain true to it as long as the cord of life shall lengthen. (Paragraph 34)
Pathos
This shows pathos because it appeals to the audiences emotions. With MLK talking of the mistreatment of family this is a emotional topic for all people. Showing the mistreatment of people of color and why they need to take action.
But when you have seen vicious mobs lynch your mothers and fathers at will and drown your sisters and brothers at whim; when you have seen hate filled policemen curse, kick and even kill your black brothers and sisters (Paragraph 14)
Logos
This shows Logos because it appeals to the audiences reason. With MLK saying that he came to Birmingham because of an organization for civil rights and they invited him to come. And showing us that this problem effects us all.
Injustice anywhere is a threat to justice everywhere. We are caught in an inescapable network of mutuality, tied in a single garment of destiny. Whatever affects one directly, affects all indirectly. Never again can we afford to live with the narrow, provincial “outside agitator” idea. (Paragraph 4)