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Me and a few friends were discussing complex exponentials and logarithms, and somehow ended up plotting the complex logarithm function using Sage, using the command complex_plot(log(x), (-5, 5), (-5, 5)). Here is the image:

enter image description here

Now, I happened to notice that I can see something vaguely resembling a white-ish colored limaçon or cardioid in this figure, which I (somewhat sloppily) trace out below in black:

enter image description here

And suddenly we all came to wonder how a limaçon of all things would be related to the complex logarithm. The theme for this post is thus pretty direct: What is this shape, and why is it there? Is it even a limaçon, or something else entirely? What relation does it have to the complex logarithm function?

According to the Sage documentation (http://doc.sagemath.org/html/en/reference/plotting/sage/plot/complex_plot.html), the brightness in the plot is the magnitude of the function, so that might be a helpful hint, althought we haven't been able to make anything of it.

Addendum after accepting answer below

This is the plot mentioned in a comment under Henning Makholms answer, made using the command complex_plot(log(x),(-6.5,6.5),(-5,5)):

enter image description here

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    If you plot the identity function with the same color scheme, do you see similarly brightish (but now straight) rays emanating from the origin in directions ±60° and 180° from the positive real axis? – 2017-01-30
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    @HenningMakholm Yes indeed, I can see those brightish lines when running complex_plot(x,(-5,5),(-5,5)) . – 2017-01-30
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    Is the phenomena you display somewhat related to [that](http://math.stackexchange.com/q/1890694) ? – 2017-01-30
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    ... taking into account that $\ln(1-z)$ is approximately $z+\dfrac{z^2}{2}$ – 2017-01-31

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If you plot the identity function with the same color scheme, you ought to see similarly brightish (but now straight) rays emanating from the origin in directions ±60° and 180° from the positive real axis.

If so, I think it is marks the arguments that are represented by hues FFFF00, FF00FF and 00FFFF, which are brighter than the intermediate hues FF0000, 00FF00, 0000FF. The cyan part is not really a loop, but simply the real interval $(0,1)$ and its immediate surroundings.

If that is the explanation, the yellow and magenta highlights do not turn smoothly into each other like you imagine -- they meet at a 120° angle at the point $-e^{\pi/\sqrt3} \approx -6.13$ on the negative real axis.

(A parametric expression for the lines is $$ z_{\text{cyan}} = e^{-t} \\ z_{\text{magenta}} = e^{-\zeta t} \\ z_{\text{yellow}} = e^{-\zeta^2 t} $$ where $\zeta = \frac12(-1 + i\sqrt3)$ is a primitive third root of unity).

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    This seems reasonable. Note that my sketch above goes slightly outside the left border of the plot. Upon remaking the plot with complex_plot(log(x), (-6.5, 6.5), (-5, 5)) instead, the bright-ish streaks do meet at an angle slightly left of (-6,0), making the shape look more like a leaf (I'll edit in a pic in my question as an addendum to show this). So this was all due to the way Sage maps angles to hues. An upvote and an accept, coming right up. – 2017-01-30