By using the laws of indices, we arrive at the laws of logarithms.

Let \(\log_a{x} = m \implies a^m = x \)

Let \(\log_a{y} = n \implies a^n = y \)

The addition law

\(a^m × a^n = xy \implies a^{m + n} = xy \)

\(\implies m + n = \log_a{xy} \)

\(\implies \boxed{\log_a{x} + \log_a{y} = \log_a{xy}} \)

This law works for three or more terms:

\(\log_a{x} + \log_a{y} + \log_a{z} + ... = \log_a{(xyz...)}\)

The subtraction law

\(\dfrac{a^m}{a^n} = \dfrac{x}{y} \implies a^{m - n} = \dfrac{x}{y} \)

\(\implies m - n = \log_a{\dfrac{x}{y}} \)

\(\implies \boxed{\log_a{x} - \log_a{y} = \log_a{\dfrac{x}{y}}} \)

The power law

Let \(\log_a{x^n} = m \implies a^m = x^n \)

Let \(\log_a{x} = y \implies a^y = x \)

\(a^m = x^n = (a^y)^n = a^{ny} \)

\(m = ny \implies \boxed{\log_a{x^n} = n\log_a{x}} \)

Change of base

This was more useful in the days before calculators had a \(\log_\square{\square}\) button. Used rarely nowadays.

\(\boxed{\log_a{b} = \dfrac{\log_n{b}}{\log_n{a}}} \)

Useful facts to remember

\(a^1 = a \iff \boxed{\log_a{a} = 1} \)

\(a^0 = 1 \iff \boxed{\log_a{1} = 0} \)