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 $5$.

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