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.
(..) ...they're as good as dead. And same as you, if you walk in that direction. Yeah, I've been dead a while and I'm getting used to it. Why don't you come with me to my base? Connor might know a way. Listen if you have a problem with the machines... ...he's definitely the guy you wanna talk to. (..)2009 Terminator Salvation
(..) How many survivors are on-site? Over. One. MAN : Copy. PILOT: Return to base, sir? -RTB? -Negative. -Take me to Command. -Sir? Command. (..)2009 Terminator Salvation