For any cardinal $\kappa$ set $$\log(\kappa) = \min\{\mu\in \kappa\cup\{\kappa\}: 2^\mu \geq \kappa\}.$$
Clearly $\log(\omega) = \omega$ and in $\textsf{GCH}$ we have $\log(\aleph\_\omega) = \aleph\_\omega$.


Is it consistent that $\log(\kappa)<\kappa$ for all uncountable cardinals $\kappa$?



Fixed points of cardinal logarithm
No. Because strong limits cardinals exist in $\sf ZFC$. These are the $\beth\_\alpha$ for a non-successor $\alpha$.


