The process of obtaining the product of two numbers via logarithms and the process of proving that the product of two circular numbers is itself also circular are complementary. The former is thick at the ends, and thin in the middle, while the latter is thin at the two ends and thick in the middle, the middle portion of both being of the same nature: separate, or split, the components and then perform an operation on those components.
Here are the schemas of the two processes.
Take each of the two items separately to the logarithmic domain.
Compute their sum.
Take that sum (via exponentiation) back to the original venue.
Remain at the original venue.
Split each item into its (canonical) complex-conjugate factorization.
Multiply, separately, the two first factors together, and the two second factors together.
Note that the result is a complex-conjugate factorization.
circular: being representable as the sum of two squares
thin: being infinitesimal, or zero
keywords: Mathematics, Number Theory, Calculus, Proof Theory, similarity of processes