Difference of Logarithms/Proof 1
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Theorem
- $\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$
Proof
\(\ds \log_b x - \log_b y\) | \(=\) | \(\ds \map {\log_b} {b^{\log_b x - \log_b y} }\) | Definition of General Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\log_b} {\frac {\paren {b^{\log_b x} } } {\paren {b^{\log_b y} } } }\) | Quotient of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\log_b} {\frac x y}\) | Definition of General Logarithm |
$\blacksquare$