Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part

Theorem
Let $z \in \C$ be a complex number.

Let $t > 0$ be wholly real.

Let $t^z$ be $t$ to the power of $z$ defined on its principal branch.

Then:


 * $\cmod {t^z} = t^{\map \Re z}$