Real Logarithm is Completely Additive

Theorem

Let $\log_b: \R_{>0} \to \R$ be the real logarithm to base $b$.

Then $\log_b$ is completely additive.

Proof

From Sum of General Logarithms:

$\log_b x + \log_b y = \log_b \left({x y}\right)$

The result follows from the definition of complete additivity.

$\blacksquare$