on 3/22/2002 10:21 AM, Ma Be at [EMAIL PROTECTED] wrote: > ... obviously whether intervals are equal or not will largely depend on > the *base* of the logarithm.
"Obvious"? It's not even true ... (... that equal logarithmic intervals depends on the base of the logarithms used). If a series has equal logarithmic intervals using logs to one base it will have equal log intervals with logs to any other base. examples, using base 10 and base e (where e = 2.718 281 828 459 045...) --------------------------------------------------------------------------- number ... log to base 10 ... log to base e 12 ... 1.079 ... 2.485 30 ... 1.478 ... 3.401 75 ... 1.875 ... 4.317 187.5 ... 2.273 ... 5.234 --------------------------------------------------------------------------- The differences between the COMMON logs (base 10) are: log 30 - log 12 = 1.478 - 1.079 = 0.399 log 75 - log 30 = 1.875 - 1.478 = 0 .397 log 187.5 - log 75 = 2.273 - 1.875 = 0.398 All differences are the same (except negligable round off error). Therefore, this sequence has equal logarithmic intervals. --------------------------------------------------------------------------- The differences between the NATURAL logs (base e) are: ln 30 - ln 12 = 3.401 - 2.485 = 0.916 ln 75 - ln 30 = 4.317 - 3.401 = 0.916 ln 187.5 - ln 75 = 5.234 - 4.317 = 0.917 This set of differences is different from the above set, but the differences in this set, too, are all the same as one another (except minor round off error). --------------------------------------------------------------------------- The set of numbers 12, 30, 75, 187.5 has equal logarithmic intervals regardless of which type of logs is used. A formal proof would require making sure that the same is true for ALL other log bases. That is done more easily by algebra rather than by repeated examples with different bases. Regards, Bill Hooper college physics teacher (retired), USA (Florida) +-+-+-+-+-+-+-+-+-+-+-+ Do It Easy, Do It Metric! +-+-+-+-+-+-+-+-+-+-+-+
