An Arithmetic and Geometric Mean Invariant
  A positive real interval, [a, b], can be partitioned into sub-intervals such that sub-interval widths divided by sub-interval "average" values remains constant. That both Arithmetic Mean and Geometric Mean "average" values produce constant ratios for the same log scale is the stated invariance proved in this short note. The continuous analog is briefly considered and shown to have similar properties. 
John Lindgren and Vibeke Libby
cs.NA math.CA