Sum of Logarithms/General Logarithm/Proof 1

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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
\(\ds \) \(=\) \(\ds \map {\log_b} {\paren {b^{\log_b x} } \paren {b^{\log_b y} } }\) Exponent Combination Laws: Product of Powers
\(\ds \) \(=\) \(\ds \map {\log_b} {x y}\) Definition of General Logarithm

$\blacksquare$