Characteristic of Decimal Logarithms

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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).

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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)