In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities, such that one quantity varies as a power of the other. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.

Maybe the following paper help you with your problem

Article Double power laws, fractals and self-similarity

Siavash Mohammadi The simplest answer is that you want the function/distribution describing fractals to be scale invariant, because the fractals themselves are scale-invariant since you see the same structure even if you zoom in or out. Simplest examples of scale-invariant functions are power laws, since they remain power laws even if their arguments undergo a scale transformation. That is why they are the favorite picks for fractals.

thank you Yousef Bahrami for the following articles.

thank you Amitava Banerjee for the explanation.

based on the articles that Yousef Bahrami sent, I think fractal systems are, pretty much by definition, self-similar which means that there is no preferred length scale. If something else depends on the length scale L as a function f(L), the argument L must have units – and no unit is better than any other – so it is "dimensionful". On the other hand, f(L) is a quantity that must have well-defined units, too, because there can't be any preferred value, either. A preferred value of f(L) would translate to a preferred value of L, and so on.