No, take Merimovich's model in which $2^\kappa=\kappa^{+3}$ for all cardinals $\kappa.$

Merimovich, Carmi A power function with a fixed finite gap everywhere. J. Symbolic Logic 72 (2007), no. 2, 361–417.

Let me add that the statement "the power function is injective" is known as weak GCH, which has important applications. See Why the weak GCH is true! .

No, take Merimovich's model in which $2^\kappa=\kappa^{+3}$ for all cardinals $\kappa.$

Merimovich, Carmi A power function with a fixed finite gap everywhere. J. Symbolic Logic 72 (2007), no. 2, 361–417.

Let me add that the statement "the power function is injective" is known as weak GCH, which has important applications. See Why the weak GCH is true! .

No, take Merimovich's model in which $2^\kappa=\kappa^{+3}$ for all cardinals $\kappa.$

Merimovich, Carmi A power function with a fixed finite gap everywhere. J. Symbolic Logic 72 (2007), no. 2, 361–417.

No, take Merimovich's model in which $2^\kappa=\kappa^{+3}$ for all cardinals $\kappa.$

Merimovich, Carmi A power function with a fixed finite gap everywhere. J. Symbolic Logic 72 (2007), no. 2, 361–417.

Let me add that the statement "the power function is injective" is known as weak GCH, which has important applications. See Why the weak GCH is true! .