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.
(..) Who would've thought he was capable of such atrocities? Oh, Slink, I hope he's okay. Oh, a survivor. Where's the rebel base? Talk! I can see your will is strong. Well, we have ways of making you talk. Where are your rebel friends now? Sid, your Pop Tarts are ready! (..)1995 Toy Story