Question

Find the value of ${\log _3}(36) - {\log _3}(4)$?

Answer 1

Hint: We will try to understand the formulas and their uses. We will use the quotient rule to change the equation in a single log. We will use the formula \[{\log _a}{x^n} = n{\log _a}x\] to minimize the log term. Then by using the property \[{\log _a}a = 1\] we will eliminate the log.

Complete answer:
We have to solve the question ${\log _3}(36) - {\log _3}(4)$
We will try to know the quotient rule.
Quotient rule states that: The log of a quotient is equal to the difference between the logs of the dividend and the divisor, according to this property. The formula is ${\log _a}(m) - {\log _a}(n) = {\log _a}\left( {\dfrac{m}{n}} \right)$
 Ex: ${\log _3}(27) - {\log _3}(9) = {\log _3}\left( {\dfrac{{27}}{9}} \right) = 1$
We have given \[{\log _a}1 = 0\]${\log _3}(36) - {\log _3}(4)$
$ = {\log _3}(36) - {\log _3}(4)$
We will us the formula ${\log _a}(m) - {\log _a}(n) = {\log _a}\left( {\dfrac{m}{n}} \right)$
$ = {\log _3}\left( {\dfrac{{36}}{4}} \right)$
$ = {\log _3}9$
We can write 9 as a square of 3 to reduce our equation.
$ = {\log _3}{3^2}$
We know that \[{\log _a}{x^n} = n{\log _a}x\] and \[{\log _a}a = 1\]
$ = 2$
Hence, the value of ${\log _3}(36) - {\log _3}(4)$ is 2

Note: The logarithmic rule will be used to rewrite the given logarithmic equation. We shall simplify the equation using several logarithmic methods like quotient rule, Product rule, Power rule etc.
Important formula are: \[{\log _a}m + {\log _a}n = {\log _a}mn\] , \[{\log _a}{m^p} = p{\log _a}m\] , \[{\log _a}1 = 0\]