Question

What is ${{\log }_{3}}9$ ?

Answer 1

Hint: We need to find the value of ${{\log }_{3}}9$ . We start to solve the problem by expressing the number 9 to the power of 3. Then, we apply the logarithmic formula given by ${{\log }_{a}}{{x}^{n}}=n{{\log }_{a}}x$ to get the desired result.

Complete step by step solution:
We are given a logarithmic expression and need to simplify the expression. We will be solving the given question using the properties of logarithms.
A logarithm, in mathematics, is the power to which a number must be raised to get some other number. It is the inverse of exponentiation.
The logarithm of base a of a positive number x is given by
$\Rightarrow {{\log }_{a}}x=b$ such that $a>0,a\ne 1$
The above expression is valid only if $x={{a}^{b}}$
The expression ${{\log }_{a}}x=b$ is read as ‘log base a of x’.
Writing the logarithmic expression given in the question as follows,
$\Rightarrow {{\log }_{3}}9$
In the above expression,
base = 3
number = 9
Now,
 Let us express the number 9 to the power of the 3.
Expressing the number 9 in terms of the number 3, we get,
$\Rightarrow 9=3\times 3$
Writing the above equation in terms of powers of 3, we get,
$\Rightarrow 9={{3}^{2}}$
Substituting the value of the number 9 in the given logarithmic equation, we get,
$\Rightarrow {{\log }_{3}}9={{\log }_{3}}{{3}^{2}}$
From the properties of logarithms, we know that ${{\log }_{a}}{{x}^{n}}=n{{\log }_{a}}x$
Here,
a = 3;
x = 3;
n = 2.
Applying the logarithmic formula to the above expression, we get,
$\Rightarrow {{\log }_{3}}9=2{{\log }_{3}}3$
From the formulae of logarithms, we know that ${{\log }_{a}}a=1$
Substituting the same, we get,
$\Rightarrow {{\log }_{3}}9=2\left( 1 \right)$
$\Rightarrow {{\log }_{3}}9=2$

Note: The result of the above expression can be verified using the rules of exponents. The result can be cross-checked as follows,
The logarithm of base a of a positive number x is given by
$\Rightarrow {{\log }_{a}}x=b$ such that $a>0,a\ne 1$
The above expression is valid only if $x={{a}^{b}}$
According to our question,
$\Rightarrow {{\log }_{3}}9=2$
Here,
x = 9;
a = 3;
b = 2.
From the above,
$\Rightarrow 9={{3}^{2}}$
LHS = RHS. The result attained is correct.