Benford's law, Zipf's law and other power laws tend to fascinate but they are a manifestation of something really simple. It's how things behave when they are evenly distributed over multiplicative operations instead of additive operations. By this I mean things are as likely to be x% bigger than x% smaller instead of the more common definition of evenly distributed (things x unit larger are as likely as x unit smaller).
Being evenly distributed over multiplication is natural for many measures. I think it is considered the most even, "maximum entropy" distribution for things that can't be negative.
The most even, maximum entropy distribution for an unbounded (-infinity to +infinity) prior is an unbounded uniform distribution but you can't apply it, for example, in the case of the size of objects since it would imply that negative sizes are as likely as positive sizes (and objects don't have negative sizes).
In general position parameters tend to be uniformly distributed over addition/subtraction whereas scale parameters tend to be evenly distributed over multiplication (uniform over log(x)). It's the reason some graphs use log axis. For Bayesians, it's a fairly straightforwards concept. See also improper priors (https://en.wikipedia.org/wiki/Prior_probability#Improper_pri...)
Being evenly distributed over multiplication is natural for many measures. I think it is considered the most even, "maximum entropy" distribution for things that can't be negative.
The most even, maximum entropy distribution for an unbounded (-infinity to +infinity) prior is an unbounded uniform distribution but you can't apply it, for example, in the case of the size of objects since it would imply that negative sizes are as likely as positive sizes (and objects don't have negative sizes).
In general position parameters tend to be uniformly distributed over addition/subtraction whereas scale parameters tend to be evenly distributed over multiplication (uniform over log(x)). It's the reason some graphs use log axis. For Bayesians, it's a fairly straightforwards concept. See also improper priors (https://en.wikipedia.org/wiki/Prior_probability#Improper_pri...)