Question

How do you evaluate \[{\log _3}(27)\]?

Answer 1

Hint: We use the property of logarithm that when the base of the log is equal to the arguments of the logarithm then the value of the logarithm equals to 1. We write the value of 27 using prime factorization with powers on prime number 3 and then use the property of log to open that value. Apply identity rules to log value in the end.
* Identity rule of logarithm states that if base is same as the argument then: \[{\log _x}(x) = 1\]
* If we have a, b and c as positive integers then \[{\log _b}(a) = c \Leftrightarrow {b^c} = a\]
* If m and n are two numbers then we can write \[\log {m^n} = n\log m\]
* Prime factorization is a process of writing a number in multiple of its factors where all factors are prime numbers.

Complete step-by-step answer:
Here we are given the function \[{\log _3}(27)\] … (1)
We will first write prime factorization of the number 27.
\[27 = 3 \times 3 \times 3\]
Since we know the law of exponents so we can collect the powers of same base
\[27 = {3^3}\]
Substitute the value of 27 in equation (1)
\[ \Rightarrow {\log _3}(27) = {\log _3}({3^3})\]
We know that \[\log {m^n} = n\log m\], then we can write
\[ \Rightarrow {\log _3}(27) = 3 \times \left[ {{{\log }_3}(3)} \right]\] … (2)
The value in the subscript is called the base of the logarithm whereas the value inside the parentheses is called the argument of the logarithm.
Here \[{\log _3}(3)\] has log base as 3 and log argument as 3
Since both base of the logarithm and argument of the logarithm are equal i.e. are equal to 3
We can say that the value of logarithm will be equal to 1
Then value of \[{\log _3}(3)\] will be equal to 1
Substitute the value of \[{\log _3}(3) = 1\] in right hand side of equation (2)
\[ \Rightarrow {\log _3}(27) = 3 \times 1\]
\[ \Rightarrow {\log _3}(27) = 3\]

\[\therefore \]The value of \[{\log _3}(27)\] is 3.

Note:
Many students make mistake of calculating the value of \[{\log _3}(27)\] by looking at the log table and directly writing the answer which is a shortcut method but students should keep in mind what if log table is not available, we can use properties of log directly here as we know 27 is 3 raised to some power.