Characteristics and mantissa of a logarithm
Consider the following.
We will take logs to the base 10 as we are dealing with common logarithms.
From the above cases, it is clear that the logarithm of a number increases as the number increases and decreases as the number decreases. It also follows that if a number lies between 0 and 10, its logarithm lies between 0 and 1. If the number lies between 10 and 100, the logarithm lies between 1 and 2, and so on.
So, the logarithm of a number is partly integral (that is, it is an integer) and partly fractional.
The integral part of a logarithm is called the characteristic and the fractional or decimal part of the logarithm is called the mantissa.
Rule for determining the characteristic of a common logarithm
Example 1:
Number 
Integral Digits
N 
Characteristic
n1 
1657 
4 
3 
208 
3 
2 
56.7 
2 
1 
81 
2 
1 
97.3614 
2 
1 
8.4123 
1 
0 
6 
1 
0 
If N, the given number, is less than 1; that is, it is a fraction or decimal and contains the first significant figures after n zeros (for example, in .0000561 the significant figures are 561 after four zeros) immediately after the decimal point, then as stated earlier it follows that a decimal fraction with no zero immediately following the decimal point lies between 10^{1} and 10^{0} (for example, 0.34). A decimal fraction with one zero following immediately after the decimal point lies between 10^{2} and 10^{1} (for example, 0.0519), etc.
or log N = (n+1) + a proper fraction
Thus, the characteristic of the logarithm of a decimal fraction N with n zeros following the decimal point is (n+1).
Example 2
with the same significant figures.
For example, the mantissa for
1) 3125, 31.25, 3.125, 0.3125 and .0003125 have the same mantissa, although their characteristics are different.
Example 3
If the mantissa of 5813 is .7644
write the logs of
a. 5813000 b. 5.813 c. 0.0005813 d. 58.13
Given that the mantissa of log 5813 = .7644
 581300
Number of integral digits = 6
Characteristic of log = n1 = 61 = 5
log 581300 = 5 + .7644
= 5.7644
 5.813
Number of integral digits = 1
Characteristic = n1 = 11 = 0
log 5.813 = 0 + .7644
= 0.7644
 0.0005813
Number of zeros after decimal point = 3
 58.13
Number of integral digits = 2
Characteristic = n1
= 21
= 1
log 58.13 = 1+ .7644
= 1.7644
Sometimes, we may obtain a logarithm that is wholly negative. In this case, we need to use arithmetical operations to write the logarithm so that the mantissa is positive.
Example 4
The result 2.17985 may be transformed by adding 1 to the integral part and +1 to the decimal part.
 2.17895 = 2 1 + (1  .7985)
= 3 + (0.82015)
=
Example 5

Write down the characteristics of the logarithms of
 3174
 625.7
 0.374
 0.00135
 1.26 * 10^{10}
 The Mantissa of log 37203 is .5705. Write down the logs of
 37.203
 3.7203
 372030000
 .000037203
 .37203
 The logarithm of 7623 is 3.8821. Write down the numbers whose logarithms are
 Given log 2 = .3010, log 3 = .4771, log 7 = .8451. Find the number of digits or zeros in
 Given log 2 = .3010 and log 3 = .4771. Find the numerical value of x in the following equations.
Answers to problems

 Number = 3174
Number of integral digits = n = 4
Characteristic = n1 = 41
= 3

Number = 625.7
Number of integral digits = n = 3
Characteristic = n1
= 31
= 2

Number = 0.374

Number = 0.00135
 Number = 1.26 * 1010
 Mantissa of log 37203 is .5705
 The logarithm of 7623 is 3.8821
 To write the number whose log is 0.8821
log 7623 = 3.8821
Let log x = 0.8821
Characteristic = 0
⇒ the number has a single digit
∴ x = 7.623
 To write the number whose log is 7.8821
log 7623 = 3.8821
Let log x = 7.8821
Characteristic = 7
⇒ number of integral digits = 7+1 = 8
∴ x = 76230000
 To write the number whose log is 1.8821
log 7623 = 3.8821
Let log x = 1.8821
Characteristic of x = 1
⇒ number of integral digits = 1+1 = 2
∴ x = 76.23

 Given log 2 = .3010, log 3 = .4771, log 7 = .8451. Find the number of digits in (42)^{42}
Solution:
 Given log 2 = .3010, log 3 = .4771, and log 7 = .8451. Find the number of digits in (81/80)^{1000}
Solution:

 Given log 2 = .3010 and log 3 = .4771. Find the numerical value of x in 3x+2 = 405
 Given log 2 = .3010, log 3 = .4771 to obtain a numerical value for x
5^{x3} = 8
5^{x3} = 2^{3}
Taking logs and using log a^{m} = m log a
(x3) log 5 = 3 log 2
xlog 5  3 log 5 = 3 log 2
x log 5 = 3 log 2+ 3 log 5