nt.number theory – 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, points, N $. For example,
$$ A_2 (8) = {2,4,8,1,6,3,2,6,4,1,2,8,2,5,6 }. $$
Similarly, define multiple sets. $ A_3 (N), A_5 (N) $ Y $ A_7 (N) $.

I can not be sure of having seen any discussion about the next question. If you do, please let me know of a reference.

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