Power Property of Logarithms

For any real number \(k\) and positive numbers \(A\) and \(b\) where \(b \ne 1\),

$$\large{\log _b}{A^k} = k \cdot {\log _b}A$$

Description: The logarithm of a number raised to an exponent can be expressed as the product of that exponent and the logarithm of the base number.

Example 1:

$${\log _7}{x^2} = 2 \cdot {\log _7}x$$

Example 2:

$${\log _2}{5^3} = 3 \cdot {\log _2}5$$