Bounds for the principal logarithm near one
= Bounds for the principal logarithm near one
{title2=$|\operatorname{Log}(1-w)|\le2|w|\quad (|w|\le1/2)$}
The <power series> $\operatorname{Log}(1-w)=-\sum_{j\ge1}w^j/j$ proves this estimate. There is no global bound $|\operatorname{Log}(1-w)|\le C|w|$, since the logarithm diverges as $w\to1$. The different estimate $\log|1-w|\le\log(1+|w|)\le|w|$ is valid wherever $w\ne1$, and is sufficient for growth bounds on <canonical products>.