# Formulas and properties of logarithms.

Definition

The **logarithm**of number b on the base a (log

_{a}b) is defined as an exponent, in which it is necessary raise number a to gain number b (The logarithm exists only at positive numbers).

_{a}b = x if and only if a

^{x}= b

## The following types of logarithms are exist

- log
_{a}b - logarithm of number b on the base a (a > 0, a ≠ 1, b > 0) - log b -
**common logarithms**(Logarithms with base 10 are called common logarithms, a = 10). - ln b -
**natural logarithms**(Logarithms with base e are called natural logarithms, a = e).

## Formulas and properties of logarithms

For any a; a > 0; a ≠ 1 and any x; y > 0.

- a
^{logab}= b - log
_{a}1 = 0 - log
_{a}a = 1 - log
_{a}(x · y) = log_{a}x + log_{a}y - log
_{a} = logx y _{a}x - log_{a}y - log
_{a} = -log1 x _{a}x - log
_{a}x^{p}= p log_{a}x - log
_{ak}x = log1 k _{a}x, for k ≠ 0 - log
_{a}x = log_{ac}x^{c} - log
_{a}x = -log _{b}xlog _{b}a**change of base formula** - log
_{a}x =1 log _{x}a

*Add the comment*