Question

How do you evaluate $ {{\log }_{9}}9 $ ?

Answer 1

Hint: We know that solving logarithmic expressions is quite easy. Here in this question, we have to find the value of $ {{\log }_{9}}9 $. You should know some properties which will be used to evaluate such expressions.
Logarithmic function is denoted as: $ {{\log }_{b}}x=n $ or $ {{b}^{n}}=x $
Power rule: $ a\log x=\log {{x}^{a}} $
 $ \Rightarrow $ ln(e) = 1(logarithm of base is 1)

Complete step by step answer:
Now, let’s solve the question.
As we know the logarithm is nothing but the power to which number must be raised to get some other values and is the most convenient way to express large numbers.
There are only two types of logarithm: common logarithm and natural logarithm.
A common logarithm is denoted as log base 10 or simply log. Whereas natural log is denoted by the natural base i.e. ‘e’ and represented as ln or loge.
There are some rules to solve logarithms. They are as follows:
Product rule: $ {{\log }_{b}}\left( mn \right)={{\log }_{b}}m+{{\log }_{b}}n $
Quotient rule: $ {{\log }_{b}}\left( \dfrac{m}{n} \right)={{\log }_{b}}m-{{\log }_{b}}n $
Power rule: $ {{\log }_{b}}\left( {{m}^{n}} \right)=n{{\log }_{b}}m $
Now, write the logarithm given in the question.
 $ \Rightarrow {{\log }_{9}}9 $
As we know that using the law of algorithms:
 $ \Rightarrow {{\log }_{b}}x=n $ or $ {{b}^{n}}=x $
Now, let $ {{\log }_{9}}9 $ = n. then by using above law we will get:
 $ \Rightarrow 9={{9}^{n}} $
As we can see that the power of 9 is one:
 $ \Rightarrow {{9}^{1}}={{9}^{n}} $
The value of n will be:
 $ \Rightarrow $ n = 1
So this is our final answer.

Note:
In the final result, we are using one more law i.e. $ {{\log }_{b}}b=1 $ . Note that you don’t need to see the value of log9 in the logarithmic table and to change the base to 10. This is the major mistake that can be made. Additionally, you should know all the power formulae of a log.