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.
(..) I'm not the fucking rat. Okay? I'm not the fucking rat. Start with you agree there is a rat. You said there is one, all right? I base most of what I do on the idea that you're pretty good at what you do. Sure. Sure, Will. You, William. What would you do? Frank, how many of these guys... (..)2006 The Departed