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I was watching some awesome zooms and came across this one:

<a href="https://www.youtube.com/v/PD2XgQOyCCk&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/PD2XgQOyCCk&rel=1&fs=1&hd=1</a>

Amazing. The description says:

So my question is... what circles? And what's the log(z) plane?? I'm pretty new to this stuff so I apologize if I missed something obvious or didn't search enough. Thanks!

As a token of my gratitude, here's a render I just did with libgmp. Hope it's not too hard on the eyes:
https://dl.dropboxusercontent.com/u/24793835/daily2.png

Thanks again

Here's an animated SVG I made some years ago that tries to explain how to convert 2x zoom levels into an exponential strip (this technique has lots of names)

http://mathr.co.uk/mandelbrot/exponential_strip.svg

The red pixel on the right is made up of a linear interpolation of 8 pixels on the left.

But you wouldn't want to calculate the exponential strip image in this way unless you had no other option - it's much less efficient than rendering an exponential strip directly and then generating 'flat' frames from it.

As ever, Mu-ency has a useful page:

http://mrob.com/pub/muency/exponentialmap.html