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.
(..) It turns out Dr. Railly was confirmed dead at the CDC by our own personnel... so you can't be her. I am Cassandra Railly. Are you aware that entering a military base under false pretense is currently punishable by death? Now, tell me who you are. Cole... Cassie. (..)2016 TvShow: 12 Monkeys Title: Lullaby Season: 2 Episode 8