= Solution
The factor $1+i$ has <modulus> $\sqrt2$ and <principal argument> $\pi/4$. Multiplication by $e^{i\pi/6}$ adds $\pi/6$ to its <complex argument>, while the real factor $1/2$ halves its <modulus>. Hence
$$
\boxed{|z|=\frac1{\sqrt2},\qquad\arg z=\frac{5\pi}{12}+2\pi k\quad(k\in\mathbb Z).}
$$
Taking the <principal logarithm> for $w$ gives
$$
w=-\frac12\ln2+i\frac{5\pi}{12}.
$$
This <complex number> lies in the second quadrant. Its <modulus> follows from the sum of squared components, and its <principal argument> follows by the second-quadrant correction:
$$
\boxed{|w|=\sqrt{\frac{(\ln2)^2}{4}+\frac{25\pi^2}{144}},\qquad
\operatorname{Arg}w=\pi-\arctan\frac{5\pi}{6\ln2}.}
$$
All arguments of this fixed $w$ add integer multiples of $2\pi$ to the displayed one.
If $\ln z$ is interpreted as the multivalued <complex logarithm>, there is instead a family $w_k=-a+ib_k$, where $a=\tfrac12\ln2>0$ and $b_k=5\pi/12+2\pi k$. Then $|w_k|=\sqrt{a^2+b_k^2}$, and the <principal argument> is $\pi-\arctan(b_k/a)$ for $b_k>0$, or $-\pi-\arctan(b_k/a)$ for $b_k<0$. This records the logarithm-branch choice that the paper leaves implicit.
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