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.
(..) l don't think l've ever seen such a flat calm. Like a millpond. Not a breath of wind. lt will make the bergs harder to see... ..with no breaking water at the base. Well, l'm off. Maintain speed and heading, Mr Lightoller. Yes, sir. lt's getting cold. (..)1997 Titanic