# Difference of Logarithms/Proof 3

## 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

$\blacksquare$