Function ${\displaystyle ~f=\ln(z)~}$ in the complex ${\displaystyle ~z~}$-plane.
Abscissa is ${\displaystyle ~\Re (z)~}$, Ordinate is ${\displaystyle ~\Im (z)~}$.
lines of constant ${\displaystyle ~\Re (f)~}$ and lines of constant ${\displaystyle ~\Im (f)~}$ are shown.
Level ${\displaystyle ~\Im (f)=-2~}$ and level ${\displaystyle ~\Im (f)=-1~}$ are shown with thick red lines.
Level ${\displaystyle ~\Im (f)=1~}$ and level ${\displaystyle ~\Im (f)=2~}$ are shown with thick blue lines.
Levels
${\displaystyle ~\Im (f)=0~}$ ,
${\displaystyle ~\Re (f)=-2~}$ ,
${\displaystyle ~\Re (f)=-1~}$ ,
${\displaystyle ~\Re (f)=0~}$ ,
${\displaystyle ~\Re (f)=1~}$ ,
${\displaystyle ~\Re (f)=2~}$ ,
are also shown with thick black lines.
Intermediate levels are shown with lines:
green is used for ${\displaystyle ~\Im (f)~={\rm {const~}}}$ ,
red is used for ${\displaystyle ~\Re (f)~={\rm {const<0~}}}$ ,
blue is used for ${\displaystyle ~\Re (f)~={\rm {const>0~}}}$ .
The cut at ${\displaystyle ~z<0~}$ is shown with multicolored line.
Gridlines are drawn with unity interval and cover the range
${\displaystyle ~-4\leq \Re (z)\leq 16~}$ ,

${\displaystyle ~-4\leq \Im (z)\leq 4~}$