Solution
= Solution
For $|w|\le1/2$, the <power series> of the <principal complex logarithm> is absolutely convergent and gives
$$
|\operatorname{Log}(1-w)|
\le\sum_{j=1}^\infty\frac{|w|^j}{j}
\le\sum_{j=1}^\infty|w|^j
=\frac{|w|}{1-|w|}
\le2|w|.
$$
Thus \b[$C_1=2$ works]. The value $1-w$ stays in the right half-plane, so there is no logarithm branch ambiguity. This is the <bounds for the principal logarithm near one> estimate used to control product tails.