According to [Whittaker, p. 258][1], a (cumulative) distribution function $G$ is called scale-invariant to base $10^t$ for some number $t>0$ if
$$\sum_{n=-\infty}^\infty [G(nt+x+y)-G(nt+y)]$$
does not depend on the value of $y$ for all real numbers $x$ and $y$. 

As for "the distribution of the frequencies of $x_m$", I think it is impossible to attach any meaning to this phrase. However, the number-theoretic density of the first (say decimal) digit of the powers of $2$ does follow Benford's law, because $2$ is not a rational power of $10$ and hence the sequence $(\log_{10}(2^n)\mod1)$ is equidistributed on $[0,1)$ -- see e.g. [Distributions known to obey Benford's law][2] and [The First Digit Problem, top paragraph on p. 525][3]. 


  [1]: https://www.jstor.org/stable/2101433?seq=1
  [2]: https://en.wikipedia.org/wiki/Benford%27s_law#Distributions_known_to_obey_Benford's_law
  [3]: https://www.jstor.org/stable/2319349?seq=1