If a, x and y are positive real numbers, where a ≠ 1, then
log_a (xy) = log_a x + log_a y.

Suppose

x = a^m and y = aⁿ,
then their equivalent logarithmic forms are
log_a x = m and log_a y = n
Consider x ÷ y.
Using the quotient property of indices, we get

Power Law of Logarithm

If a and x are positive real numbers, where a ≠ 1, and k is any real number, then

log_a x^k = k log_a x.

The next law of logarithms, tells us how logarithms to different bases are related.

Change of Base Law of Logarithm

If x, a and b are positive real numbers, where a ≠ 1 and b ≠ 1, then

This law enables us to calculate the logarithm of a number to any base from a calculator which calculates the logarithm of a number to only one base.