Sum of Logarithms/General Logarithm/Proof 1

Theorem

Let $x, y, b \in \R$ be strictly positive real numbers such that $b > 1$.

Then:

$\log_b x + \log_b y = \map {\log_b} {x y}$

where $\log_b$ denotes the logarithm to base $b$.

Proof

\(\ds \log_b x + \log_b y\)

\(=\)

\(\ds \map {\log_b} {b^{\log_b x + \log_b y} }\)

Definition of General Logarithm

\(\ds \)

\(=\)

\(\ds \map {\log_b} {\paren {b^{\log_b x} } \paren {b^{\log_b y} } }\)

Product of Powers

\(\ds \)

\(=\)

\(\ds \map {\log_b} {x y}\)

Definition of General Logarithm

$\blacksquare$