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CAT 2025 Lesson : Logarithm - Characteristic & mantissa

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4. Characteristic and Mantissa



Characteristic is the integer part, before the decimal point, of a logarithm of a number.
Mantissa is the fractional part, after the decimal point, of a logarithm of a number.

In log10200\mathrm{log} _{10} {200} == 2.3012.301 == 2+0.3012 + 0.301
The characteristic and mantissa of log10_{10} 2 are 2 and 0.301 respectively.

In the case of common logarithm (base 10), characteristics and mantissa follow certain patterns. Note the following

log102\mathrm{log} _{10} {2} == 0.3010.301
log1020\mathrm{log} _{10} {20} == 1.3011.301
log10200\mathrm{log} _{10} {200} == 2.3012.301
log102000\mathrm{log} _{10} {2000} == 3.3013.301
log100.2\mathrm{log} _{10} {0.2} == 1‾.30103\overline{1}.30103 == −1+0.301 -1 + 0.301 == −0.699-0.699
log100.0002\mathrm{log} _{10} {0.0002} == 4‾.30103\overline{4}.30103 == −4+0.301 -4 + 0.301 == −3.699-3.699

Note: Mantissa is a positive number. So, if the logarithm of a number is negative, then the mantissa should be made positive. In the case of log100.0002\mathrm{log} _{10} {0.0002} = −3.699-3.699 == −4+0.301-4 + 0.301, characteristic and mantissa are −4-4 or 4‾\overline{4} and 0.3010.301 respectively. Characteristic of negative numbers are typically represented by an overline.

1) If the characteristic of a common logarithm is a positive number, say xx, then the number of digits of the number to the left of the decimal point is x+1x + 1. Where xx is an integer, if log10x\mathrm{log} _{10} {x} == 4.664.66 , then xx is a 5-digit number (characteristic + 1).

2) If the characteristic of a common logarithm is a negative number, say xx, then the number of zeroes to the right of the decimal point and before the first non-zero digit is x−1x - 1. If log10x\mathrm{log} _{10} {x} =−2.347= -2.347 == −3+0.653-3 + 0.653, then number of zeroes between the decimal point and the first non-zero number is 3−1=23 - 1 = 2.

3) If log10a=b\mathrm{log} _{10} {a} = b, then log(a×10n)\mathrm{log} {(a \times 10^n)} = nn + bb

Example 9

Given log 5=0.699\mathrm{log} \ 5 = 0.699, what are the values of log 5000\mathrm{log} \ 5000, log 0.05\mathrm{log} \ 0.05 and log (5×10−3.4)\mathrm{log} \ (5 \times 10^{-3.4})

Solution

As the base is not given, these are considered as common logarithms with base 1010.

log 5000=\mathrm{log} \ 5000 = log(5×103)=\mathrm{log} (5 \times 10^{3}) = 0.699+3=0.699 + 3 = 3.699\boldsymbol{3.699}
log 0.05\mathrm{log} \ 0.05 =log(5×10−2)= \mathrm{log} (5 \times 10^{-2}) =0.699−2== 0.699 - 2 = −1.301\boldsymbol{-1.301}
log 5×10−3.4\mathrm{log} \ 5 \times 10^{-3.4} =0.699−3.4= 0.699 - 3.4 =−2.701= \boldsymbol{-2.701}

Alternatively, log 5×10−3.4\mathrm{log} \ 5 \times 10^{-3.4} =log 5+log 10−3.4== \mathrm{log} \ 5 + \mathrm{log} \ 10^{- 3.4} = log 5+(−3.4)log 10\mathrm{log} \ 5 + (-3.4) \mathrm{log} \ 10
=0.699−3.4= 0.699 - 3.4 =−2.701= \boldsymbol{-2.701}

Answer: 3.6993.699; −1.301-1.301; −2.701-2.701


Example 10

If the logarithm of an integer approximated to 22 decimal places is 16.8916.89, then how many digits does the integer have?

Solution

As the characteristic of the number is 1616, the number of digits in the number =16+1=17= 16 + 1 = 17

Answer: 1717


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