Show that the set of powers of ten is countably infinite

My question asks: An $x \in \mathbb{N}$ is called a power of ten if there exists a $y \in \mathbb{N}$ such that $10^y = x$. Show that the set of powers of ten is countably infinite.

I understand that for the set of powers to be countably infinite, there must exist a bijection between the set and $\mathbb{N}$.

$y\mapsto 10^y$ is your desired bijection: It is onto by definition, and readily verified to be one-to-one.