Decimal logarithms, that is, in base 10, have characteristics in common. Note the possible location of the numbers in relation to the base 10 powers:
100 < 2,56 < 101
101 < 32,5 < 102
102 < 600,37 < 103
We can define the above situation as follows: 10 c ≤ x < 10 c + 1. For every positive real number x there is an integer c. Based on this idea, we can establish that:
10 ç ≤ x < 10 c + 1
log 10 ç ≤ log x < log 10 c + 1
c * log 10 ≤ log x < c + 1 * log 10
c ≤ log x < c + 1
log x = c + m, where 0 ≤ m < 1.
We conclude that the decimal logarithm of a number x is the sum of an integer c with a decimal m less than 1, where the decimal m is called the mantissa. Watch:
log 620
10² < 620 < 10³ → log10² < log 620 < log10³ → 2 * log 10 < log 620 < 3 * log 10
2 < log 620 < 3, so we have the integer part of the logarithm of the number will be equal to 2.
To prove this property, just use a scientific calculator, through the keylog. Enter the number, in case 620 and press the log key, note that we will have as a result the decimal number 2.792391..., which is composed of the integer part equal to 2 and decimal 0.7922391... (mantissa).
In determining the 0.0879 log we have to:
10–2 < log 0.0879 < 10 –1 → log 10 –2 < log 0.0879 < log 10 –1
–2 * log 10 < log 0.0879 < –1 * log 10 → –2 < log 0.0879 < –1
The integer part of the logarithm of the number will be equal to –1.
Using the calculator we have:
log 0.0879 → –1.0560
Another option in determining the logarithm characteristic of a numeral is related to two situations: x > 1 and 0 < x < 1.
Situation: x > 1
When x > 1, the characteristic of the log is equal to the number of digits of the integer part subtracted from 1.
log 1230 → 4 – 1 = 3 (characteristic 3)
log 125 → 3 – 1 = 2 (characteristic 2)
12500 → 5 – 1 = 4 (characteristic 4)
Situation: 0 < x < 1
In this case, the characteristic will be determined through the symmetry of the number of zeros that precede the first significant digit.
log 0.032 → feature 2
log 0.00000785 → feature 6
log 0.0025 → feature 3
by Mark Noah
Graduated in Mathematics
Brazil School Team
Logarithm - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/caracteristica-dos-logaritmos-decimais.htm
