Question

What is ${{\log }_{3}}27$?

Answer 1

Hint: In this question we have been given with a logarithmic expression which we have to simplify and get the value. We will first convert the given number in the form of exponent. We will then use the property of logarithm that ${{\log }_{p}}{{a}^{b}}=b{{\log }_{p}}a$, we will then use the property of logarithm that ${{\log }_{a}}a=1$ and substitute it and simplify to get the required solution.

Complete step by step solution:
We have the expression given to us as:
$\Rightarrow {{\log }_{3}}27$
We can write $27$ as ${{3}^{3}}$ therefore, on substituting, we get:
$\Rightarrow {{\log }_{3}}{{3}^{3}}$
Now on using the property of logarithm that ${{\log }_{p}}{{a}^{b}}=b{{\log }_{p}}a$, we get the expression as:
$\Rightarrow 3{{\log }_{3}}3$
Now on using the property of logarithm that ${{\log }_{a}}a=1$, we get the expression as:
$\Rightarrow 3\times 1$
On multiplying the terms, we get:
$\Rightarrow 3$, which is the required value of the given expression.
Therefore, we can write:
$\Rightarrow {{\log }_{3}}27=3$, which is the required solution.

Note: It is to be noted that the logarithm we are using has the base $3$, the base is the number to which the log value has to be raised to, to get the original term. This is also called the antilog of the number which is the logical reverse of taking a log.
The most commonly used bases in logarithm are $10$ and $e$ which has a value of approximate $2.713...$
Logarithm is used to simplify a mathematical expression, it converts multiplication to addition, division to subtraction and exponents to multiplication. The various properties of logarithm should be remembered while doing these types of sums.