Exponent to which another number called the

*base of the logarithm*must be raised to obtain a given number.The integer part of a logarithm is called the

*characteristic*, and its fractional part is called the*mantissa*.- A
**common logarithm**is a logarithm to the base 10. - A
**natural logarithm**– or Napierian logarithm – is a logarithm to the base e.

### Notation

The logarithm of the number *a* to the base *b* is written as log* _{b}*(

*a*), where

*b*is a positive number not equal to 1.

### Properties

- The logarithm of 1 is equal to 0: log
_{b}(1) = 0. - The logarithm of a product
*xy*is equal to the sum of the logarithms of its factors*x*and*y*: log_{b}(*xy*) = log_{b}(*x*) + log_{b}(*y*), if*x*> 0 and*y*>0. - The logarithm of a quotient \(\frac{x}{y}\) is the difference of the logarithms of the dividend and of the divisor: log
_{b}\(\frac{x}{y}\) = log_{b}(*x*) – log_{b}(*y*), if*x*> 0 and*y*> 0. - The logarithm of a power
*x*^{y}is equal to the product of the exponent*y*and the logarithm of*x*to the base*b*: log_{b}(*x*^{y}) =*y*log_{b}(*x*), if*x*> 0. - The logarithm of a root \(\sqrt[x]{y}\) is equal to the logarithm of the number
*y*, whose root is sought, divided by the exponent*x*: log_{b}(\(\sqrt[x]{y}\)) = \(\frac{1}{x}\) log_{b}(*y*), if*y*≥ 2 and*y*> 0.

### Example

If the logarithm to the base 10 of 100, written as log_{10}(100), is 2, then: 100 = 10².

Therefore, the **mantissa** (fractional part) is 0, and the **characteristic** (integer part) is 2.

The characteristic also gives the *order of magnitude* of 2000.

In this case, the order of magnitude is 10².

It can be noted that the expression log_{10}(100) = 2 is equivalent to the expression 102 = 100.