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.
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.
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.
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.
You'll want to make sure that the end of the fade out is at a "zero crossing" in the waveform.
Half a second is pretty fast. You might just want to extend the amount of time, unless it must be that fast. Generally 2 or 3 seconds is more natural.
More on timing, it should really be with the beat rate of the music, and end at a natural point in the rhythm. Try getting the BPM of the song (this can be calculated roughly), and fading out over an interval equal to a whole or half note in that timing.
You might also try slowing down the playback speed while you're fading out. This will give a more natural vinyl record or magnetic tape sounding stop/pause. Linearly reduce playback speed while logarithmically reducing volume over the period of 1 second.
If you're just looking to get a clean sound sound when pausing or stopping playback then there's no need to fade at all - just find a zero-crossing point and stop there (or more realistically just fill the rest of that final buffer with silence). Fading out when the user expects the sound to stop immediately will sound unnatural, as you've noticed, because the result is decoupled from the action.
The reason for stopping at a zero-crossing point is that zero is the steady state value while the audio is stopped, so the transition between the two states is seamless. If you stop playback when the last sample's amplitude is large then you are effectively introducing transients into the audio from the point of view of the audio hardware when it reconstructs the analogue signal, which will be audible as pops and/or clicks.
Another approach is to fade to zero very fast (~< 10mS), which effectively achieves the same thing as the zero-crossing technique.