Difference of Logarithms/Proof 3

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Theorem

$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$


Proof

\(\displaystyle \map {\log_b} {\frac x y} + \log_b y\) \(=\) \(\displaystyle \map {\log_b} {\frac x y \times y}\) Sum of Logarithms
\(\displaystyle \) \(=\) \(\displaystyle \log_b x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map {\log_b} {\frac x y}\) \(=\) \(\displaystyle \log_b x - \log_b y\) subtracting $\log_b y$ from both sides

$\blacksquare$


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