“I Have a Dream” by Martin Luther King Jr. mainly appeals to ethos and pathos, though there are also occasional appeals to logos. The speaker uses these forms of appeal to encourage the audience to support the Civil Rights Movement by illustrating why this movement is necessary and what it will achieve.
King appeals to reason in the speech by mentioning facts and using logical arguments which support his views on racial discrimination.
For example, he lists several real situations in which African Americans are discriminated against, using these facts to support his claim that the problem is significant, and that African Americans have a legitimate reason to be angry and protest:
We can never be satisfied as long as the Negro is the victim of the unspeakable horrors of police brutality. We can never be satisfied as long as our bodies, heavy with the fatigue of travel, cannot gain lodging in the motels of the highways and the hotels of the cities. (ll. 82-87)
Note that this quotation also appeals to pathos, both because of the injustices it describes and because of the strong choice of words (e.g. "unspeakable horrors").
Furthermore, King points out that discrimination against specific groups is logically inconsistent with the message of the Declaration of Independence, which argued for the unalienable rights of all men.
This note was a promise that all men, yes, black men as well as white men, would be guaranteed the ‘unalienable Rights’ of ‘Life, Liberty and the pursuit of Happines...