# Logarithm Calculator | lg(x) Calculator

## Logarithm

With our logarithm calculator, you can calculate the logarithm of a number, multiply logarithms, divide, exponentiate and take the root from the logarithm. Also, you will find all formulas and definitions to your calculation just input the required parameters and get the result immediately.

The logarithm of x with respect to base b, where b > 0, b \ne 1, is the exponent by which b must be raised to yield the number x.

Denoted as \log_{b}{x} and read as the logarithm of the number x to base b.

It follows from the definition that the equation y = \log_{b}{x} is equivalent to the equation b^y = x.

Example, \log_{2}{8} = 3 because 2^3 = 8.

The numbers x and b are often real numbers, but there are also complex logarithms.

Logarithms have unique properties that have determined their wide application with the possible simplification of complex calculations.

## Real logarithm

The expression \log_{b}{x} only makes sense if b > 0, a > 0, a \ne 1.

The following logarithms are widely used:

- Natural:\log_{e}{x} or \ln{x}, based on the Euler number (e - irrational mathematical constant ≈ 2.71828).
- Common: \log_{10}{x} or \lg{x}, base 10.
- Binary: \log_{2}{x}, base 2.

They are widely used, in computer science, discrete mathematical divisions, etc.

## Basic logarithmic identity

After the definition of the logarithm comes the basic logarithmic identity. b^{\log_{b}{x}} = b If \log_{b}{x} = \log_{b}{y} , then b^{\log_{b}{x}} = b^{\log_{b}{y}} , which implies that x = y .