I noticed the other day while computing consecutive powers of $2$ that for $n \geq 1$, the numbers in the ones place of the values of $2^n$ repeat every 4 terms $(2, 4, 8, 6,\ldots)$. In the tens place, we get a repetition every 20 terms for $n \geq 4$ (the first term with a value in the tens place). In the hundreds place, a repetition every 100 for $n \geq 7$. The values in the thousands place repeat every 500 terms for $n \geq 10$.
Thus, for $2^n$, the pattern in the $10^k$ place will repeat every $4\cdot5^k$ terms for $n \geq 1+3k$
What is this called? Is there a way to generalize for every $x^n$?
When you have a finite set $S$, a map $f:\ S\to S$ and an initial value $a_0\in S$ then the sequence $(a_n)_{n\geq0}$ recursively defined by $$a_{n+1}:=f(a_n)\qquad(n\geq0)\tag{1}$$ will eventually become periodic: After at most $|S|$ steps the recursion will produce a number which you have seen before, and from then on the procedure will repeat periodically.
Now given an $m>1$ the possible remainders of integers modulo $m>0$ form such a finite set $S$, and multiplication by $2$, i.e. $$\tilde f:\quad{\mathbb N}\to{\mathbb N}, \quad k\mapsto 2k\ ,$$ acts exactly like $(1)$ on these remainders. The remainder modulo $10^r$ of a natural number $k$ is plainly visible in the last $r$ digits of the decimal representation of $k$. It follows that beginning with an arbitrary $a_0$ the last $r$ digits of the numbers $a_n:=2^n a_0$ will eventually become periodic with a period length $\leq 10^r$.