<p>I have experimented with a sigmoid and logarithmic fade out for volume over a period of about half a second to cushion pause and stop and prevent popping noises in my music applications.</p>

<p>However neither of these sound "natural". And by this I mean, they sound botched. Like an amateur engineer was in charge of the sound decks.</p>

<p>I know the ear is logarithmic when it comes to volumes, or at least, twice as much power does not mean twice as loud. Is there a magic formula for volume fading? Thanks.</p>

I spent many of my younger years mixing music recordings, live concerts and being a DJ for my school's radio station and the one thing I can tell you is that where you fade is also important.
Fading in on an intro or out during the end of a song sounds pretty natural as long as there are no vocals, but some of these computerized radio stations will fade ANYWHERE in a song to make the next commercial break ... I don't think there's a way to make that sound good.
In any case, I'll also answer the question you asked ... the logarithmic attenuation used for adjusting audio levels is generally referred to as "audio taper".  Here's an excellent article that describes the physiology of human hearing in relation to the electronics we now use for our entertainment.  See:
http://tangentsoft.net/audio/atten.html
.