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 )

[b]uThe addition law[/u]/b

(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...)})

[b]uThe subtraction law[/u]/b

(\dfrac{a^m}{an} = \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}}} )

[b]uThe power law[/u]/b

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}} )

[b]uChange of base[/u]/b

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}}} )

[b]uUseful facts to remember[/u]/b

(a¹ = a \iff \boxed{\log_a{a} = 1} )

(a⁰ = 1 \iff \boxed{\log_a{1} = 0} )

[b]uLaws of logarithms[/u]/b

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

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

(\log_a{x^n} = n\log_a{x} )

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

[b]uUseful log facts[/u]/b

(\log_a{a} = 1 )

(\log_a{1} = 0 )