Can you "undo" a powerset of an infinite set?

The boring answer is just that it doesn't make sense to talk about the "$\log_2$" of an infinite cardinal.

One problem is the one you discovered: some cardinals, like $\aleph_0$, are not equal to $2^\kappa$ for any cardinal $\kappa$. So there's no obvious way to define the logarithm of such a cardinal. To put it another way, the exponential function $\kappa\mapsto 2^\kappa$ is not surjective on infinite cardinals.

Even worse, it's consistent with ZFC that the exponential function fails to be injective on infinite cardinals. For example, we could have $2^{\aleph_0}=2^{\aleph_1}$. What should $\log_2$ of this cardinal be?