Logarithms have numerous applications in everyday life, Physics and Chemistry use logarithmic functions in phenomena in which numbers acquire very large values, making them smaller, facilitating calculations and the construction of graphics. The handling of logarithms requires some properties that are fundamental for its development. Look:
Logarithm Product Ownership
If we find a logarithm like: logThe (x * y) we must solve it by adding the logarithm of x to base a and the logarithm of y to base a.
logThe (x * y) = logThe x + logThe y
Example:
log2 (32 * 16) = log232+ log216 = 5 + 4 = 9
Logarithm Quotient Properties
If the logarithm is of type logThex/y, we must solve it by subtracting the logarithm of the numerator in base a from the log of the denominator also in base a.
logThex/y = logThex - logThey
Example:
log5 (625/125) = log5625 - log5125 = 4 – 3 = 1
Log power property
When a logarithm is raised to an exponent, on the next pass that exponent will multiply the result of that logarithm, here's how:
logThexm = m*logThex
Example:
log3812 = 2*log381 = 2 * 4 = 8
Root property of a logarithm
This property is based on another, which is studied in the property of rooting, it says the following:
no√xm = x m/n
This property is applied in the logarithm when:
logTheno√xm = logThe x m
no
→ m • logThex
no
Example:
log23√162 = log2162/3 = 2 • log216 = 2 • 4 = 8
3 3 3
Base Change Ownership
There are situations in which we will need to use a logarithm table or a scientific calculator to determine the logarithm of a number. But for that we must work the problem in order to establish the logarithm in base 10, because the tables and the calculators operate under these conditions, for this we use the base change property, which consists of the following definition:
logBa = logçThe
logçB
Example
log58 = log 8 = 0,90309 = 1,292
log 5 0.69898
by Mark Noah
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/propriedades-operatorias-dos-logaritmos.htm
