Cardinal exponentation

Question

Let $\lambda, \kappa$ be infinite cardinals. Does it follow from the inequality $\kappa\lt2^\lambda$ that $2^\kappa\lt2^{2^\lambda}$?

Answer 1

No.

Suppose $2^{\aleph_0}=\aleph_2$ and $2^{\aleph_1}=2^{\aleph_2}=\aleph_3$.

Let $\lambda=\aleph_0$ and $\kappa=\aleph_1$, now we have $2^\kappa=2^{2^\lambda}$.