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.