You've 'based' your opinion 'on' faulty information. = Your opinion is 'based on' faulty information. [=the information that you used to form your opinion is wrong]
The company's 'base' is in London. = The company's 'base of operations' is (in) London. [=the company's main offices are in London]
Although I disagreed with the book's theoretical 'base' [='basis, foundation'], I couldn't really find fault with its logic.
(..) So it's a goddamn tragedy. So, let's have a moment of silence. Someone's trying to block the way. Gotta be the Saviors. Look. I think that's their base over there. Yeah, that's it. Must be trying to make it hard to get to them. We gotta keep going. We'll move them, and then (..)2017 TvShow: The Walking Dead Title: Rock in the Road Season: 7 Episode 9