What are some examples of logos in the "I Have a Dream" speech?

Dr. Martin Luther King, Jr. uses rhetorical appeal—or what Aristotle calls “modes for persuasion” —in his famous I Have a Dream speech: ethos (appealing to ethics or credibility), logos (appealing to logic), and pathos (appealing to emotion). Perhaps most critical to the success of his speech is his use of logos.
In fact, his second sentence is an undeniable fact: “Five score years ago a great American in whose symbolic shadow we stand today signed the Emancipation Proclamation.” He follows this immediately with an example of logos:

“But one hundred years later, the Negro still is not free; one hundred years later, the life of the Negro is still sadly crippled by the manacles of segregation and the chains of discrimination; one hundred years later, the Negro lives on a lonely island of poverty in the midst of a vast ocean of material prosperity; one hundred years later, the Negro is still languished in the corners of American society and finds himself in exile in his own land.”

His use of repetition functions to “drive the point home” in a logical manner; no one can deny the truth of this statement. Through this logic, he is able to elicit an emotional response from the crowd.
He also uses logos to speak to the personal, individual truths of many audience members:

“I am not unmindful that some of you have come here out of great trials and tribulations. Some of you have come fresh from narrow jail cells. And some of you have come from areas where your quest — quest for freedom left you battered by the storms of persecution and staggered by the winds of police brutality.”

King phrases this truthful, logical statement in a way that resonates with the victims of police and prison brutality, as well as those who sympathize with the aforementioned struggle, creating comradery as an emotional response to logic. Similarly, he later extends the sentiment of comradery to the white community:

“The marvelous new militancy, which has engulfed the Negro community, must not lead us to a distrust of all white people. For many of our white brothers, as evidenced by their presence here today, have come to realize that their destiny is tied up with our destiny. And they have come to realize that their freedom is inextricably bound to our freedom. We cannot walk alone. And as we walk, we must make the pledge that we shall always march ahead. We cannot turn back.”

Again, his use of repetition aids his rhetoric, as he reminds the crowd that—emotions aside—white allies exist, again eliciting a feeling of comrodery and an emotional response to a logical statement.