Third person plural pronoun is THEY.
If you want to create a possessive pronoun out of that one, that would be THEIRS.
Possessive pronouns, as the name itself says, determine some kind of possession. For example:
Is this books yours? - No, it's theirs.
The message is understood but the sentence is not well written. You should write "Logos, that is the appeal of logic, consists of persuation backed up with actual observation, data, and facts". While the word Logos can be paraphrased as 'the appeal of logic', it is necessary to clarify the function of Logos in a text. With that purpose in mind, it is neccesary to separate the ideas of persuation from the paraphrase.
The author's argument about life in the mountains would be that life was emotional and beautiful. According to the text, it states, "while the heights and the meadow lands were turning gold under the rising sun, which was just appearing above the topmost peaks." Another piece of evidence is: "<span>The roaring in the trees always stirred a strong emotion within her and seemed to draw her irresistibly to them." </span>This supports the author's argument that life in the mountains was actually very enchanting.
The best revision of the statement "Children seem to like me, and I don't mind hanging out with them. It's pretty easy to keep them amused" is:
Children seem to like me, and it's pretty easy to work with them.
<em>This revision deletes the phrase "I don't mind hanging out with them", that phrase is not really necessary since it was already stated that children liked the person. It is logical that the person doesn't mind hanging out with someone who likes him/her. Also, the phrase "it's pretty easy to keep them amused" suggests that working with them is easy. At the end of the day, the key to work with children is being able to keep them amused, in other words, getting and keeping their attention. </em>
