I think 1 is special, in any base, because it's the first digit used when a new digit gets added. If you're talking about quantities that vary easily by, say, thousands, once it crosses the 10,000 threshold the first digit changes much more slowly.
There's more to think about here for sure, though.
[Not regarding you comment or this response]
As I have understood, the Law applies only to a logarithmic scale [1 to 2, 2 to 3, 3 to 4, ... to ...]. Look at this pattern graph:
Try it on a set of numbers in base 10, then convert them to base 16 and check the percentages you get. They still follow the same pattern, but of course there are more slots and the individual percentages are lower because of that.