Real Logarithm is Completely Additive
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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.
$\blacksquare$