Logarithmic Fades:

This type of fading is often a favorite of engineers because of the natural sound it produces. Here's what Autumn Fell sounds like with a logarithmic fade. Can you tell the difference? See if you can by comparing it to the slower linear fade. You can hear the music in the log fade a lot longer.

af6.gif (11416 bytes) speaker1.gif (364 bytes)

Can you figure out the mathematics behind the logarithmic fade? How would we graph an example?

Let's do some reviewing and exploring.

If we graph 

 , log x, base 2  and  log x, base 3/2 ...

threelog.gif (3072 bytes)
(Graphing window: x on [-2.58, 13.98], y on [-2, 5.1])

Notice that the smaller the base (3/22 and 10 in our example), the steeper the graph.

Now, how can we get our logs to flip over the y-axis like they are in our fades? By using f(-x) instead of f(x)!

Let's check by graphing  

log2negx.gif (2474 bytes)
(Graphing window: x on [-13.98, 2.38], y on [-2, 5.1])

Getting back to our logarithmic fades... When the engineer decides how fast the song should fade, the program adjusts the base of the logarithm!

Music is just applied mathematics...
A good musician is an applied mathematician!

Eddie Van Halen? Math geek? You be the judge!