<s>[INST] Frequency of digits in powers of $2, 3, 5$ and $7$ For a fixed integer $N\in\mathbb{N}$ consider the multi-set $A_2(N)$ of decimal digits of $2^n$, for $n=1,2,\dots,N$. For example,
$$A_2(8)=\{2,4,8,1,6,3,2,6,4,1,2,8,2,5,6\}.$$
Similarly, define the multi-sets $A_3(N), A_5(N)$ and $A_7(N)$.
I can't be sure if I have seen any discussion of the below question. If you do, please do let me know of a reference.

QUESTION. For $N$ large, is it true that the most frequent digit in $A_x(N)$ is $x$, where $x\in\{2,3,5,7\}$?

 [/INST] It is not an answer but some numercal data for $A_2(N)$. Here digits are ordered according to their frecuences ($2$, $4$ and $6$ look like most frequent):
$$
\begin{array}{rl}
N=1000: & 2, 1, 4, 6, 8, 9, 5, 3, 0, 7\\
N=2000: & 2, 1, 6, 4, 8, 5, 3, 9, 7, 0 \\
N=3000: & 6, 2, 1, 4, 3, 8, 0, 5, 7, 9 \\
N=5000: & 2, 0, 4, 1, 8, 6, 5, 3, 7, 9 \\
N=6000: & 4, 8, 2, 1, 9, 7, 0, 6, 5, 3 \\
N=7000: & 4, 9, 8, 7, 1, 2, 3, 6, 5, 0 \\
N=8000: & 4, 8, 7, 1, 9, 2, 3, 6, 5, 0 \\
N=10000: & 4, 8, 7, 1, 2, 9, 6, 3, 5, 0 \\
N=15000: & 4, 2, 8, 1, 6, 7, 3, 0, 5, 9 \\
N=20000: & 6, 4, 3, 8, 2, 1, 7, 9, 0, 5 \\
N=25000: & 6, 4, 2, 3, 8, 1, 9, 7, 5, 0 \\
N=30000: & 6, 9, 4, 3, 2, 8, 1, 0, 7, 5 \\
\end{array}$$
</s>