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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\log_b} {\paren {b^{\log_b x} } \paren {b^{\log_b y} } }\) | Product of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\log_b} {x y}\) | Definition of General Logarithm |
$\blacksquare$