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

Latest Math Problems

Email subscription

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.