According to Whittaker, p. 258, 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 $y$. Whittaker's Theorem 4 characterizes the scale-invariant distribution functions in terms of their characteristic functions.
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 and The First Digit Problem, top paragraph on p. 525.