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

$\blacksquare$