Question

Write the characteristics of the following.
(1) \[\log 27.91\]
(2) \[\log 0.02871\]
(3) \[\log 0.000987\]
(4) \[\log 2475\]

Answer 1

Hint: Here we are given the logarithmic values and we are asked to find the characteristics of the following values. First of all, we see the given value lies in which interval. Then we will use the concept of characteristics and mantissa i.e., the common logarithm of a positive number consists of two parts. One part is integral which may be zero or any integer known as characteristics and the other part is non-negative decimal which is known as the mantissa.
\[ \Rightarrow \log x = \] characteristics \[ + \] mantissa.

Complete step by step answer:
Here, we need to find the characteristics of the logarithmic value.
Take, \[\log 27.91\]
Here, \[27.91\] lies between \[10\] and \[100\]
We choose \[10\] and \[100\] because we know the logarithmic values of these terms.
i.e., \[10 < 27.91 < 100\]
Taking logarithm on both sides,
\[ \Rightarrow \log 10 < \log 27.91 < \log 100\]
\[ \Rightarrow 1 < \log 27.91 < 2\]
[Since \[\log 10 = 1\] and \[\log 100 = 2\] ]
Therefore, the logarithm of a number between \[10\] and \[100\] lies between \[1\] and \[2\]
i.e., \[\log 27.91 = 1 + \] a positive decimal part
Now, if we compare it with \[\log x = \] characteristics \[ + \] mantissa.
We get, characteristics \[ = 1\]
Hence, the characteristics of \[\log 27.91\] is \[1\]
Now similarly we will find all remaining logarithms.
Take \[\log 0.02871\]
Here, \[0.02871\] lies between \[.1\] and \[.01\]
i.e., \[.01 < 27.91 < .1\]
Taking logarithm on both sides,
\[ \Rightarrow \log .01 < \log 0.02871 < \log .1\]
\[ \Rightarrow - 2 < \log 0.02871 < - 1\]
[Since \[\log .1 = - 1\] and \[\log .01 = - 2\] ]
Therefore, the logarithm of a number between \[.01\] and \[.1\] lies between \[ - 2\] and \[ - 1\]
i.e., \[\log 0.02871 = - 2 + \] a positive decimal part
Now, if we compare it with \[\log x = \] characteristics \[ + \] mantissa.
We get, characteristics \[ = - 2\]
Hence, the characteristics of \[\log 0.02871\] is \[ - 2\]

Now, take \[\log 0.000987\]
Here, \[0.000987\] lies between \[.0001\] and \[.001\]
i.e., \[.0001 < 0.000987 < .001\]
Taking logarithm on both sides,
\[\log .0001 < \log 0.000987 < \log .001\]
\[ - 4 < \log 0.000987 < - 3\]
[Since \[\log .0001 = - 4\] and \[\log .001 = - 3\] ]
Therefore, the logarithm of a number between \[.0001\] and \[.001\] lies between \[ - 4\] and \[ - 3\]
i.e., \[\log 0.000987 = - 4 + \] a positive decimal part
Now, if we compare it with \[\log x = \] characteristics \[ + \] mantissa.
We get, characteristics \[ = - 4\]
Hence, the characteristics of \[\log 0.000987\] is \[ - 4\]

Now, take \[\log 2475\]
Here, \[2475\] lies between \[1000\] and \[10000\]
We choose \[1000\] and \[10000\] because we know the logarithmic values of these terms.
i.e., \[1000 < 2475 < 10000\]
Taking logarithm on both sides,
\[ \Rightarrow \log 1000 < \log 2475 < \log 10000\]
\[ \Rightarrow 3 < \log 2475 < 4\]
[Since \[\log 1000 = 3\] and \[\log 10000 = 4\] ]
Therefore, the logarithm of a number between \[1000\] and \[10000\] lies between \[3\] and \[4\]
i.e., \[\log 2475 = 3 + \] a positive decimal part
Now, if we compare it with \[\log x = \] characteristics \[ + \] mantissa.
We get, characteristics \[ = 3\]
Hence, the characteristics of \[\log 2475\] is \[3\]

Note:
To solve this question we can also use the concept i.e., the characteristics of the logarithm of a number greater than \[1\] is positive and is one less than the number of digits in an integral part of the number.
Like in the part (1), the number is \[\log 27.91\] which is greater than \[1\] and number of digits in the integral part is \[2\]
So, the characteristics of the logarithm will be one less than the number of digits n the integral part.
i.e., \[2 - 1 = 1\]
Hence, the characteristics of \[\log 27.91\] is \[1\]
Similarly in the part (4), the number is \[\log 2475\] which is greater than \[1\] and number of digits in the integral part is \[4\]
So, the characteristics of the logarithm will be one less than the number of digits in the integral part.
i.e., \[4 - 1 = 3\]
Hence, the characteristics of \[\log 2475\] is \[3\]

And the characteristic of the logarithm of a positive number less than one is negative and is numerically greater by one than the number of zeros between the decimal sign and the first significant figure of the number.
Like in the part (2), the number is \[\log 0.02871\] which is less than \[1\] and the number of zeros between decimal part and first significant figure is \[1\]
So, the characteristics of the logarithm will be negative and numerically greater by one than the number of zeros
i.e., \[ - \left( {1 + 1} \right) = - 2\]
Hence, the characteristics of \[\log 0.02871\] is \[ - 2\]
Similarly in the part (4) the number is \[\log 0.000987\] which is less than \[1\] and the number of zeros between decimal part and first significant figure is \[3\]
So, the characteristics of the logarithm will be negative and numerically greater by one than the number of zeros
i.e., \[ - \left( {3 + 1} \right) = - 4\]
Hence, the characteristics of \[\log 0.000987\] is \[ - 4\]