= Solution
The logarithm's argument is $\sqrt3/2+i/2=e^{i\pi/6}$, with <complex modulus> one. The principal <complex logarithm> is therefore $\ln1+i\pi/6=i\pi/6$, giving
$$
\boxed{\Re\operatorname{Log}z=0,\qquad\Im\operatorname{Log}z=\frac\pi6.}
$$
If $\ln z$ denotes all logarithm values rather than the principal branch, they are $i(\pi/6+2\pi k)$ for $k\in\mathbb Z$: the <real part> remains zero and the <imaginary part> changes by integer multiples of $2\pi$. The principal value is the conventional single-valued answer.
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