The cardinal equation $\kappa^{\aleph_0}=2^\kappa$ is satisfied by $\kappa=\aleph_0$.

It is also satisfied by any $\kappa$ for which $MA(\kappa)$ holds.

Under $GCH$, the equation is satisfied by $\kappa$ if and only if $cof(\kappa)=\aleph_0$.

So my question is:

Is it consistent with $ZFC$ that the only solution to $\kappa^{\aleph_0}=2^\kappa$ is $\kappa=\aleph_0$?

I´m sorry if this is too basic, but I just don´t see it.