"Malcolm Stewart" wrote in
message ...
"Bruce" wrote in message
...
...... The 12-34 has very even ratios of 10 to 15%.
12-14-16-18-20-23-26-30-34

15.4% = ln(14/12)
13.4%
11.4%
10.5%
14.0%
12.3%
14.3%
12.5%

-Bruce

I'm new to this. Why the need to use natural logs in the above
calculation?
What's wrong with simple ratios between adjacent pairs?

Cheers
--
M Stewart
Milton Keynes, UK
www.megalith.freeserve.co.uk/oddimage.htm




Q: Why the need to use natural logs in the above calculation?
A: The beauty of symmetry:

18/20 = 9/10 = 0.90 = -10% change
20/18 = 10/9 = 1.111 = 11.1% change.
ln(20/18) = minus ln(18/20) = 0.10536 = 10.5%

While I can do the first two calculations w/o computer assitance (and hence
while riding) I perfer the symmetric choice. That way when I downshift from
the 18 to the 20 it's the same change as when I upshift from the 20 to the
18.

Note the key word here is 'prefer'. Choose which method you prefer.

-Bruce
pure mathematician since age 7.