The integral part of a common logarithm is called characteristic.

The logarithm of a quantity is written as integral and fractional forms. The integral part is usually positive, negative or zero. It is called characteristic in logarithms.

$\log{(Q)} \,=\, C + \log{(m)}$

In this case, the literal $C$ represents characteristic and the $\log{m}$ is called mantissa.

If characteristic is negative, then just write magnitude of the characteristic and display a bar over it. It’s mainly to avoid the addition with mantissa and it doesn’t impact the value of the logarithm of a quantity.

$651983$ is a quantity and let’s find characteristic of the logarithm of this number.

$\log{(651983)} \,=\, \log{(6.51983 \times {10}^5)}$

Use product rule of logarithms to express log of product of two or more quantities as sum of their logs.

$\implies$ $\log{(651983)}$ $\,=\,$ $\log{(6.51983)} + \log{({10}^5)}$

Now, use power rule of logarithm to write the logarithm of exponential term as product of two quantities.

$\implies$ $\log{(651983)}$ $\,=\,$ $\log{(6.51983)} + 5\log{(10)}$

The base of common logarithm is $10$. So, the logarithm of $10$ is one as per logarithm of base rule.

$\implies$ $\log{(651983)}$ $\,=\,$ $\log{(6.51983)} + 5 \times 1$

$\implies$ $\log{(651983)}$ $\,=\,$ $\log{(6.51983)} + 5$

$\implies$ $\log{(651983)}$ $\,=\,$ $5+\log{(6.51983)}$

The number $5$ is integrated with log of a quantity and this integral part of logarithm is called characteristic. Therefore, the characteristic of $\log{(651983)}$ is $5$.

Latest Math Topics

Aug 31, 2024

Aug 07, 2024

Jul 24, 2024

Dec 13, 2023

Latest Math Problems

Oct 22, 2024

Oct 17, 2024

Sep 04, 2024

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved