In this episode we provide the missing uniqueness part for our construction of general powers. More precisely, we will show that given any continuous function that satisfies the power law is actually a power. The technique to obtain this is by successively checking cases of increasing complexity: if the function satisfies the power law it behaves like a power for natural numbers, for integers, for rationals (using the uniqueness of the n-th root of non-negative numbers), and, finally, using continuity and density, for reals.