.
Here, we consider the inverse of the mapping, 
, namely,
By the choice of capital letter here, we mean the
principal branch of the multi-valued logarithm function, for which
Figure 10: Pre-image of the polar region of Figure 9.
As a check, consider the mapping of Figure 10 where a rectangular region mapped into a polar region. By using PolarMap in Mathematica with the polar region of that figure, we should generate the pre-image, namely, the rectangular domain that produced that image. That image is shown in Figure 10.
Figure 11: Mapping by Logz of a line just above the x-axis.
A more interesting plot is generated by the image of a straight line above the
axis in the z-plane, as in Figure 11. The line in the z-plane
was 
At z = 10 +.02i,, Log 
with imaginary part nearly zero
because 
As z passes over the origin with x moving from
positive to negative values, 
changes from 0 to 
. The nearest
point to the origin is z = .02i where 
As x moves
to more negative values, moves towards while |z| moves back
towards 10. Hence, the upper part of the curve here is the image of the part of
the line where x ;SPMlt; 0.
Figure 12: Mapping of the box with vertices, (-2,.2), (2,.2), (2,4.2),
(-2,4.2) by the function, Logz.
This helps us understand the image of a rectangular region in Figure
12. Each line x = constant has as image a U-shaped curve,
symmetric about the line 
The family of orthogonal curves are the
images of the lines y = constant. In particular, the straight line and curve
passing through 
are the images of the lines,
x = 0 (the horizontal line) and y = 1 (the curve).
The reader should verify.