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That works. Well done.
Use the polar form of a + bi = R*e^it where
R = sqrt(a^2 + b^2) and t = arctan(b/a)
So ln(a + bi) = ln(R*e^it)
= ln(R) + it
= 1/2*ln(a^2 + b^2) + i* arctan(b/a)
Of course, in the complex numbers, any integer multiple - 2npi - added to t also works. What we both have done here is find the principal value when n= 0.
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