Defined, ethos is:
...is...based on a Greek word and denotes the guiding beliefs or ideals that characterize a community, a nation or an ideology...
"Ethos" provides a sense of a culture's norms with regard to standards that guide those norms, but there is more:
Ethos...refers to the trustworthiness or credibility of the writer or speaker. Ethos is often conveyed through tone and style of the message...
As an example, I would look to the ancient story of Beowulf, which is about a heroic man who puts his life in harm's way to save others from a monster—Grendel. Ethos is easy to spot in this story when we look to the tone. Here the storyteller quotes Beowulf:
I reckon myself to be in the ready for grim deeds of war, and in no way weaker than Grendel. For this reason will I not give his life to the sleep of death with a sword, although I could. He has no skill to strike me with sword or hew through shield, mighty though he may be in his horrific feats...Let the wise God, the holy Lord, decree success on whichever side seems right to Him!
In terms of "ethos," this passage extols Beowulf for his bravery, his sense of fair play (he refuses the advantage a sword would give him), and his humility—looking to God to decide who will prevail. So while ethos may generally reflect a "community's" guiding principles, it also has to do with how we view the writer as a credible source; and, we learn that it may not be always about what the writer says outright, but what we discover when we "read between the lines" to analyze the "tone an style" used. With "ethos," we may find ourselves searching for meaning rather than having it delivered directly to us.
...represents an appeal to the audience's emotions.
In this case, the audience is being "manipulated" to feel a certain way based upon the content of something written. Referring again to the character of Beowulf, we may feel differently about our hero after reading John Gardner's book Grendel—here the story is told from the monster's viewpoint. "Pathos" is present in that the monster is presented as a lonely outcast who feels he is wrongly despised—he is the victim: attacked by humankind and cursed as a descendant of Cain—the first murderer. The reader finds it harder to see Grendel as a clear villain until the story's end, when "all the facts are in."
With this said, the following definition makes sense in terms of the example above:
[Pathos] is often associated with emotional appeal. But a better equivalent might be "appeal to the audience's sympathies and imagination."
This is much the case in Grendel.
Logos is defined in the following way:
[It] refers to the internal consistency of the message--the clarity of the claim, the logic of its reasons, and the effectiveness of its supporting evidence. The impact of logos on an audience is sometimes called the argument's logical appeal.
In the stories of Beowulf vs. Grendel, "logos" reveals whether our initial emotional response to feel sorry for Grendel is borne out by his overall actions. We initially feel empathy for him; "logos" refers to whether our response is warranted or not—in this story, Grendel is a monster; we can't forgive him.
These three elements control how our reader will respond to the message we hope to share in the essay, but using them effectively requires great skill.