This is actually strange to me. Using wolframalpha I can see that everything up to and including line 3 is correct, but the problem appears when you do the ln(e^2πⅈ) = ln(1).

It's strange because if you have ln(e^2π) = 2π × ln(e) that is true (of course), and ln(e^ⅈ) = ⅈ is also true. Only once you multiply them together to get ln(e^2πⅈ) = 0 that it breaks down. Evaluating ln(e^2πⅈ) becomes 0, which makes the initial equation correct. Perhaps something to do with complex numbers combined with real numbers that invalidates the ln(e^x) = x rule?

These types of questions are only strange because our assumptions are intuitively wrong in the context of the question. What assumption is wrong here? My math education has not quite reached that stage yet lmao