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.
(..) none wed the second but who killed the first Wormwood The instances that second marriage move are base respects of thrift, but none of love A second time I kill my husband dead when second husband kisses me in bed I do believe you think what now you speak, This world is not for aye, (..)2015 Hamlet
(..) - It waves me forth again, I'll follow it - What if it tempt you towards the flood, my lord? Or to the dreadful summit of the cliff that overlaps his base into the sea'? And there assumes some other horrible form which might deprive you of your reason... ...and draw you into madness. Think of it (..)2015 Hamlet