However, it still puzzles me that logarithm of some physical quantity has no units. Example, let's assume we have a collection of values of the distance between two cities. In set A, distances are expressed in km, while in set B distances are expressed in m.
If I apply a logarithmic transformation to both sets and compute the average and standard deviation I get two different values for the first parameter and the same for the latter.
The latter makes sense because I'm subracting logarithms and that is equal to a division operation so I get dimensionless values for the standard deviation.
For the average value I get different values. Assuming that the logarithmic transformation returns dimensionless values in principle I could sum the average values of sets A and B and get a number, however this doesn't make sense since sets A and B are not expressed in the same scale of length unit. Therefore one can argue that you must still somehow keep track of dimensions when you do logarithmic transformations or you might end up summing apples and oranges.
What is your take on the above?