Before one analyzes Prose's essay for her use of logos, it is necessary to understand her thesis. Prose argues that both the low quality of books selected for study in American schools and the dumbed-down methods for teaching classic books don't inspire in students a desire for reading literature.
Logos is the technique of persuading an audience through the application of logic and reason to the argument.
In "I Know Why the Caged Bird Cannot Read," Prose makes a claim of value: the purpose of studying literature is to learn and appreciate "the powers of language [to] connect us, directly and intimately, with the hearts and souls of others" and not "to make us examine ourselves." To logically support her claim, Prose cites, for example, the folly of a lesson plan when teaching The Diary of Anne Frank that asks students to gather the items they'd want to take with them if they found themselves in the situation of Jews moved to ghettos and concentration camps.
For the part of her argument about schools opting for novels of dubious quality, Prose first establishes that "lightweight and mediocre" novels are those that reduce human behaviors and reactions to them as "good and bad" and then cites that that kind of thinking produces the simplistic observation that in To Kill a Mockingbird, "Scout and Atticus are good, their bigoted neighbors are bad."
Appeals to logos are arguments that are meant to persuade through logic. This is in contrast to those arguments taht appeal to pathos, trying to win emotional sympathy. In this essay by Prose, there are many places in which she tries to win us over with logic and reasoned argument.
One example of this is where Prose gathers the reading lists from some eighty or so high schools around the country. She notes which works are read most often, thus providing factual data to support her contention about what sorts of things high school students are reading.
Another example of this comes when Prose tells us about the purpose for which literature is being taught. She cites, for example, the San Francisco Board of Education's requirement that literature be taught in a way that will reflect the diversity of the community. By telling us this, she lays the basis for her argument that literature is being taught in the wrong way and that this sort of teaching is turning students off to serious literature.