# Logarithm

A logarithm is a shorthand way of expressing exponentional notation.

## Introductory

The general form for a logarithm can be expressed as $\log_x y=z$ which means $x^z=y$. We would read this as "The logarithm of y base x is z". We have $3^3=27$. To express this in Logarithmic notation, we would write it as $\log_3 27=3$. When a logarithm has no base, it is assumed to be base 10.

## Logarithmic Properties

These hold for all logarithms.

• $\log_a b^n=n\log_a b$
• $\log_a b+ \log_a c=\log_a bc$
• $\log_a b-\log_a c=\log_a \frac{b}{c}$
• $(\log_a b)(\log_c d)= (\log_a d)(\log_c b)$
• $\frac{\log_a b}{\log_a c}=\log_c b$
• $\log_a^n b^n=\log_a b$