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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 the logarithm of the dividend (numerator) subtracted by the logarithm of the divisor (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:

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

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