I was wondering if this language is context-free:
$L = \{ x \in \{ 0, 1 \}^* : |x| = 2^n $ for some natural number n $\}$
I know that this language is not regular because it fails the pumping lemma for regular languages but that does not necessarily mean it is not context-free. I'm not sure whether to use the pumping lemma for context-free languages to show that this is not context free or to provide context-free grammar to show that it is context free.
I've tried creating a context-free grammar to generate this language but ran into trouble which makes me believe that this language is not context-free, but I am still unsure.
If someone could point me in the right direction that would be greatly appreciated.