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 $A_7(N)$.

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