Why and how mathematically can it be proved ?
From my analysis :
X-axis will be $n$ value.
Y-axis will be the n-th sum of digits of $x^n$ form.
On the current image, where $x=2$, the black line is a regression line with $a$ slope.
If we get slopes $a$ for $x \in (1, 2, \dots, 100)$, then the curve looks:
From further analysis, it seems that this curve is $\log(x) * \alpha$ where $\alpha$ is some constant.
The number of digits of $2^n$ in base $10$ is $\lceil n \log_{10}2\rceil \approx 0.30103 n$ for $n \gt 1$. When $n$ is large you expect the average digit to be $4.5$. The first few and the last few may not be evenly distributed, but for large $n$ the effect will be negligible so we expect the sum of digits of $2^n$ to be about $1.3546n$