Sum of Logarithms/General Logarithm/Proof 2
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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
\(\ds \log_b x + \log_b y\) | \(=\) | \(\ds \frac {\ln x} {\ln b} + \frac {\ln y} {\ln b}\) | Change of Base of Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\ln x + \ln y} {\ln b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \ln {x y} } {\ln b}\) | Sum of Logarithms: Proof for Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\log_b} {x y}\) | Change of Base of Logarithm |
$\blacksquare$