Complex log mapping maps radii to horizontal lines and circles to vertical lines. 
The principal branch w=Log⁡z{\displaystyle w=\operatorname {Log} z}, viewed as a mapping from C−R≤0{\displaystyle \mathbb {C} -\mathbb {R} _{\leq 0}} to the horizontal strip defined by |Im⁡z|&lt;π{\displaystyle \left|\operatorname {Im} z\right|&lt;\pi }, has the following properties, which are direct consequences of the formula in terms of polar form:

Circles[1] in the z-plane centered at 0 are mapped to vertical segments in the w-plane connecting a−πi{\displaystyle a-\pi i} to a+πi{\displaystyle a+\pi i}, where a{\displaystyle a} is the real log of the radius of the circle.
Rays emanating from 0 in the z-plane are mapped to horizontal lines in the w-plane.
Each circle and ray in the z-plane as above meet at a right angle. Their images under Log are a vertical segment and a horizontal line (respectively) in the w-plane, and these too meet at a right angle. This is an illustration of the conformal property of Log.
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