Product Property of Logarithms

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

$${\log _b}AB = {\log _b}A + {\log _b}B$$

Description: The logarithm of a product is equal to the sum of the logarithms of its factors.

Examples 1:

$${\log _2}3x = {\log _2}3 + {\log _2}x$$

Example 2:

$$\eqalign{
   {\log _5}\left( {24} \right) &= {\log _5}\left( {3 \cdot 8} \right)  \cr 
  &  = {\log _5}3 + {\log _5}8 \cr} $$

Example 3:

$$\eqalign{
   {\log _4}\left( {30} \right) &= {\log _4}\left( {2 \cdot 3 \cdot 5} \right)  \cr 
  &  = {\log _4}2 + {\log _4}3 + {\log _4}5 \cr} $$