Language to Generate Powers of 2 Using a Language Containing Decimal Numbers
L is not context-free
Periodicity for context-free languages of bounded growth: Let $w_n$ be the number of words of length $n$ in a given regular language. If the sequence $w_1, w_2, \cdots$ is bounded (bounded growth), then it is eventually periodic.
Proof: This follows from the fact that every context-free language of bounded growth is a union of paired loops, which is the theorem 2.1 in the paper On a conjecture about slender
context-free languages by Lucian Ilie, 1994.

Let $w_n$ be the number of words of length $n$ in $L$. $w_n=\#\{k\in\Bbb N\mid n-1\le k\log_{10}2 \lt n\}$. Since $\log_{10}2$ is irrational, the sequence $w_1, w_2, \cdots$ cannot be eventually periodic. So $L$ is not context-free. Hence, it is not regular, either.
As rici pointed out, it is not difficult to prove that $L$ is not context free by applying the pumping lemma to any word in $L$ that is long enough. The above more conceptual proof is presented in the hope that it might raise more interest and more questions.
$L$ is context-sensitive
According to rici's comment, we can write a context-sensitive grammar which multiplies a decimal number by two. Use marker tokens which move right to left over the number; you'll need two of them because of the possibility of carry. As with many other cases of context-sensitive grammars, the actual construction might be apt to get lengthy and tricky, sometimes even debatable.
Another conceptual way to see $L$ is context-sensitive is using linear bounded automaton (LBA) that accepts a context-sensitive language. It is easy to see that we can divide by 2 repeatedly using a linear bounded automaton. More specifically, we can design a LBA such that given an input of $S m E$, where $m$ is a sequence of decimal digits and where $S$ and $E$ are start marker and end marker respectively, it will output $S1\square\square\cdots\square E$, where $\square$ stands for the special blank symbol if and only if $m$ is the decimal representation of some power of 2.
An exercise
Let language $L_U$ over $\Sigma=\{0,1,2,3,4,5,6,7,8,9\}$ be the language of words which are positive integers whose prime factors belong to a finite set $U$ of prime numbers when they are treated as decimal numbers. Show that $L_U$ is context-sensitive but not context-free.