am i doing this right?

[ilikephysics]

Find the value of ln(-e), (-1)^i, and (2i)^1+i in rectangular x+iy form, using the principal value of the logarithm.

This is what I'm doing for each one:

ln(-e)= e^ln(-e)

(-1)^i= e^(i)(ln(-1))

(2i)^1+i= e^(1+i)ln(2i)

If this is right, where do I go from here?

[HallsofIvy]

Well, each statement is TRUE but do they help at all?

One reason the problem specifically says "in rectangular form" is that you will want to convert to "polar form" to do the calculations- then convert back.

If you write a complex number z= x+iy in "polar form" as
r eiθ, then ln(z)= ln(r)+ iθ Notice that in the first for, r eiθ, adding or subracting π to θ will not change z but will change ln(z). The point of "principal value of the logarithm" is to specify θ: it must be between 0 and 2π.
In particular, -e= e ei(π). That is, r= e and θ= iπ. ln(-e)= ln(e)+iπ= 1+ iπ

For (-1)i, use the fact that (eiθ)i= e(i*i)θ= e-θ. -1= eiπ so -1i= e-π.