# Base-Invariance

Working on exercise a, you saw that changing the base only changes the formula from log base 10 to log base whatever you are working with. But it is more subtle, i.e. the procedures in scale-invariance cannot be repeated, because the set of numbers with leading digit 1 (to 1.99...) in base 10 cannot be translated to base 7 as numbers with some other leading digit (whereas scaling by say 4.5 means looking at numbers with leading digit(s) 45 to 90. In the language of probability, not all probability can be computed and the ones that can be is called measurable).

What we can compare is ..., base 0.1, base 10, base 100, base 1000, ... (or base b^n, where b>1), because leading digit 1 to 2 in base 10 is leading digit 1 to 2 or 10 to 20 in base 100. Rewriting it as 10^0 to 10^(log 2) in base 10 and 100^0 to 100^(log 2 /2) or 100^(1/2) to 100^[(1+log 2)/2, we see the right definition of a base-invariant probability would be something like

where b is the base.

Using base-invariance will not give Benford's law because the number 1 is special as changing the base will leave 1 unchanged. Removing 1, we will get Benford's law (after some analysis which we won't do. For the interested reader, see "Base-Invariance Implies Benford's Law").