Complex Logarithms

Following up on my post here, I came across another such fun stuff here. Although it is already typed on that page, im retyping it, with the chance of trying to elaborate a bit more. The argument goes as follows:

1. We know that e^(i*theta) = cos(theta) + i sin(theta)
2. now, if theta = pi radians, then, we know that e^(i Pi) = cos(Pi) + i sin (Pi)
3. Now, cos(Pi rad) = cos (180 deg) = -1, and sin(Pi) = 0
4. so, e^(i Pi) = -1
5. now, square on both sides, so we get:
6. e^(2 i Pi) = 1
7. now take ln on both sides, so as to get
8. 2 i Pi = 0

clearly this is absurd, so something strange must be going on when taking a log of a complex number (step 7). and it turns out that yes, indeed this is a tricky situation, and this leads us to the topic of "Complex Logarithms". Essentially what is going on here is the following (as Wikipedia explains): A logarithm of a complex number essentially has "infinite" answers. (kind of like "aliasing"). Will revisit this page soon.