Difference of Logarithms/Proof 1

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

$\blacksquare$

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.