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.
(..) There's a sheep trail that takes you there. It's a mile out of town headed west, not far. Just through a small pass here near the base of the hills. You'll pass a church and just follow the sheep trail. -Where's Standard buying up? -Here and here. -Your family name is Sunday, yes? (..)2007 There Will Be Blood