Logarithm Properties - Product & Quotient Rule

The most important logarithm properties are product and quotient rules. Many logarithm formulas are derived from these rules.

Formula

(Product Rule) ${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$

(Quotient Rule) ${\log }_a\frac{x}{y}={\log }_{a}x-{\log }_{a}y$

In particular

(Power rule) ${\log }_a{x}^n=n{\log }_{a}x$

The logarithm of the multiplication of real values is the total of the logarithms of the values.

And the logarithm of the division of real values is the difference of the logarithms of the values.

Proof

Reviewing exponent expressions help us to prove the above.

$a^m\times a^n=a^{m+n}$

Let $x$ be $a^m$ and $y$ be $a^n$.

$x\times y=a^{m+n}$

And then

${\log }_a(x\times y)=m+n$

${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$

So the rules of logarithm are essentially ones of exponential.