Log of Complex number using both positive and negative angles

Question

I came across this question. Determine in polar form $\ln(−4−i3)$,

Using the Euler's I converted $(−4−i3)$ to:

i) $5e^{i3.79}$ using counterclockwise angle

ii) $5e^{-i2.5}$ using clockwise angle

Then performing ln gives me two different polar answers. Apparently the book uses i) for the answer. Where am I doing wrong?

Answer 1

What's the problem? Every non-zero complex number has infinitely many logarithms. Your two approaches lead to distinct logarithms. That's all.