Probability of First Digits in a Sequence of Powers

A more general theorem holds where $2$ can be replaced with any other integer, except for $10;$ and there is even a more general formulation. Given an $r-digit$ number $N.$ We may inquire about appearance of $N$ as the first digits in the decimal representation of the powers of $.2.$ From here, the problem can be stated for an irrational $\alpha,$ not a rational power of $10:$

For a given $r-digit$ number $N,$ and an irrational $\alpha,$ as above, what is the probability that the decimal representation of $\alpha^n$ begins with $N.$

Define, for $x \gt 0,$ $\mu(x)$ as the mantissa of $\log x,$ that is,

$\mu(x) = \log x - \lfloor\log x\rfloor =\log x (\text{mod}\;1).$

Let $S=\{\alpha^n\}_{i=1}^{\infty}.$ Then $\mu(S)$ is dense in the open interval $(0,1)$ which is proved with a relatively standard invocation of the Pigeonhole principle, see also another example.

Due to the equidistribution theorem, the probability that the decimal representation of $\alpha^n$ begins with $N$ is the probability that $\mu(\alpha^n)$ lies in the interval $I=\left[\mu(N),\mu (N+1)\right)$ whose length is

$\mu (N+1)-\mu (N)=\log (N+1)-\log N=\log\left(1+\frac{1}{N}\right).$

Note that that probability is independent of $\alpha.$

Note also that, from the density of $\mu(S)$ is in $(0,1)$ it follows that, for any $N,$ there are infinitely many powers of $\alpha$ whose decimal representation starts with $N.$

This is problem 50 from the Canadian Crux Mathematicorum (v 1, 1975). The problem was proposed by John Thomas with the following comment: I found the following fascinating two-part problem in Martin Eisen's Introduction to Mathematical Probability Theory (Prentice-Hall 1969). Information as to its origin and history would be appreciated.

The solution above is a shorten version of that by Léo Sauvé.

The problem is an extension of an earlier discussion.