The decimal logarithm system was proposed by Henry Briggs with the purpose of adapting the logarithms to the decimal numbering system. In the case of the decimal system, only powers of 10 with integer exponents have integer logarithms.
Examples:
log 1 = 0
log 10 = 1
log 100 = 2
log 1000 = 3
log 10,000 = 4
log 100,000 = 5
log 1 000 000 = 6
In this way, the position of the logarithms of numbers can be discovered as follows:
The logarithms of numbers between 1 and 10 have results between 0 and 1, those included between 10 and 100 are between 1 and 2, those between 100 and 1000 are between 2 and 3 and so on against.
Examples
Check which whole numbers are between:
a) log 120
100 < 120 < 1000 → 10² < 120 < 10³ → log 10² < log 120 < log 10³ → 2 < log 120 < 3
The log of 120 is between 2 and 3
Using the scientific calculator, we have log 120 = 2.079181246047624827722505692704
b) log 1 342
1000 < 1342 < 10000 → 10³ < 1342 < 104 → log 10³ < log 1342 < log 104 → 3 < log 1342 < 4
The log of 1342 is between 3 and 4
log 1342 = 3.1277525158329732698496873797248
c) log 21
10 < 21 < 100 → 10 < 21 < 10² → log 10 < log 21 < log 10² → 1 < log 21 < 2
The log of 21 is between 1 and 2
log 21 = 1.3222192947339192680072441618478
d) log 12 326
10 000 < 12 326 < 100 000 → 104 < 12 326 < 105 → log 104 < log 12 326 < log 105
4 < log 12 326 < 5
log 12 326 = 4,09082163394656573599272585104
by Mark Noah
Graduated in Mathematics
Brazil School Team
Logarithms - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/sistema-logaritmos-decimais.htm