Quotient Property of Logarithms

For positive numbers \(A\), \(B\), and \(b\) but \(b \ne 1\),

$${\log _b}{A \over B} = {\log _b}A – {\log _b}B$$

Description: The logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator.

Example 1:

$${\log _3}{x \over 2} = {\log _3}x – {\log _3}2$$

Example 2:

$${\log _7}{3 \over {10}} = {\log _7}3 – {\log _7}10$$

Example 3:

$$\eqalign{
 {\log _6}7 &= {\log _6}{{14} \over 2}  \cr 
  &  = {\log _6}14 – {\log _6}2 \cr} $$