# 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

Factoring: Some special cases
Formulas and properties of exponents
Formulas and properties of nth root
Formulas and properties of logarithms
Formulas and properties of arithmetic sequence
Formulas and properties of geometrical sequence
Trigonometry formulasDerivative formulas
Integrals table

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