Knowledge is to cut, divide, separate (etymologically).
Logic is to collect (etymologically).
Knowledge requires logic to arrive at truth.

I think the above demonstrates a rule that is previously ignored which would prevent the tool of logic from leading someone off course. The user uses a tool but if the rules of how to use it aren't known, then it can be misused. Logic will work on a soundly constructed origin.

But when using a=b, that violates the law of identity. It's not explicitly shown, but that's what the above states. 'ab' can only equal 'a^2' if a=b. Constructing a false setup and using logic on a false setup will produce a false result. The logic isn't the problem. In the end, with the false result, one can recognize its falsity through logic that collects and compares the data for consistency and non-contradiction.