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.
(..) ...and made love under that oak... ...and I asked and she said yes. Promise me, Red. If you ever get out... ...find that spot. At the base of that wall, there's a rock that has no earthly business in Maine. Piece of black, volcanic glass. Something's buried under it I want you to have. What, Andy? (..)1994 The Shawshank Redemption