Question

How do you find the exact value of \[{{\log }_{3}}\left( 9 \right)\]?

Answer 1

Hint: Convert the argument of the given logarithmic expression, i.e., 9 into the exponential form having base 3. Now, apply the logarithmic formula given as: - \[\log {{a}^{m}}=m\log a\] to simplify the expression. Now, use the identity: - \[{{\log }_{n}}n=1\] to get the answer. Here, ‘n’ denotes the same argument and base of the logarithmic function and n > 0, \[n\ne 1\].

Complete step by step answer:
Here, we have been provided with the logarithmic expression: - \[{{\log }_{3}}9\] and we are asked to find its exact value.
Now, the given logarithmic expression is neither a common log nor a natural log but it has a base 3. As we can see that the argument of the log is also a multiple of 3 (square of 3), so assuming the value of this logarithmic expression as ‘E’, we have,
\[\Rightarrow E={{\log }_{3}}9\]
Converting the argument, i.e., 9, into the exponential form, we can write the expression as: -
\[\Rightarrow E={{\log }_{3}}\left( {{3}^{2}} \right)\]
Using the logarithmic identity: - \[\log {{a}^{m}}=m\log a\], we get,
\[\Rightarrow E=2{{\log }_{3}}3\]
Now, since the base and argument of the logarithmic expression is same in the above relation, so applying the formula \[{{\log }_{n}}n=1\], where n > 0 and \[n\ne 1\], we get,
\[\begin{align}
  & \Rightarrow E=2\times 1 \\
 & \Rightarrow E=2 \\
\end{align}\]

Hence, the exact value of the logarithmic expression \[{{\log }_{3}}9\] is 2.

Note: One must remember that common log has base 10 and natural log has base e where \[e\simeq 2.71\]. Natural log is denoted by ln. Here, in the above question the base of the log was 3 so it was neither a common log nor a natural log. Note that logarithm is the inverse process of exponentiation so we can also solve the above question by converting the logarithmic form of the equation into its exponential form. What we will do is we will use the basic definition of logarithm to write: - if \[{{\log }_{3}}9=x\] then \[{{3}^{x}}=9\]. In the next step we will write \[9={{3}^{2}}\Rightarrow {{3}^{x}}={{3}^{2}}\] and then remove the base 3 from both the sides by equating the exponent. We will solve for the value of x to get the answer.