Another characterization of the cofinality?

Question

Is it true that $cf(\kappa)=\min \{\lambda:\ \kappa^{\lambda}>\kappa\}$?

$cf(\kappa)$ is certainly $\geq$ than that minimum since $\kappa^{cf(\kappa)}>\kappa$, but I don't know how to tackle the inverse inequality. What's worse, I'm not even sure if my hypothesis is true :) Is it?

(I'm actually fighting to prove that this $\min$ is a regular cardinal, and my idea was that in could be equal to the cofinality).

Answer 1

It's consistent with $\mathsf{ZFC}$ that $2^\omega=2^{\omega_1}=\omega_2$, in which case $\omega_1^\omega=\omega_2>\omega_1$, even though $\omega<\omega_1=\operatorname{cf}\omega_1$. Thus, it's at least consistently false.

Answer 2

For infinite cardinal $k,$ it's also consistent with ZFC that it is true because of Konig's Theorem (aka Konig's Lemma): $k^{cf(k)}>k,$ and because GCH implies that $k^l=k$ for $0<l<cf(k).$