# Decimal Logarithm. Characteristic and Mantissa of Decimal Logarithm

If base of logarithm equasls 10, then logarithm is called decimal. Instead of record log_(10)(x) record lg(x) is used.

In particular, for decimal logarithms we have that color(green)(10^(lg(a))=a) and color(blue)(lg(10^n)=n).

Suppose that positive number a is represented in standard form: a=a_1*10^n, where 1<=a_1<=10, n in ZZ (n is number exponent of a). Let's take decimal logarithm of number a and use properties of logarithms.

We have that lg(a)=lg(a_1*10^n)=lg(a_1)+lg(10^n)=lg(a_1)+n.

So, color(red)(lg(a)=lg(a_1)+n).

Since 1<=a_1<=10 then lg(1)<=lg(a_1)<=lg(10), i.e. 0<=lg(a_1)<=1.

Therefore, from equality lg(a)=lg(a_1)+n it follows that n is the greatest integer number, that is less or equal lg(a), in other words n is integer part of lg(a): n=[lg(a)].

Summand lg(a) is fractional part of lg(a), i.e. lg(a_1)={lg(a)}.

Integer part of number lg(a) i.e. number exponent of a is called characteristic of lg(a), fractional part of number lg(a) is called mantissa of lg(a).

Following fact is true: if we multiply number a>0 by 10^k, where k is integer number, then mantissa of logarithm will not change, in other words lg(a) and lg(a*10^k) have same mantissas.