Question

Is $\log 100$ rational or irrational? Justify your answer.

Answer 1

Hint: In the above question, we have asked whether $\log 100$ is rational or not. Now, to solve this problem, we will use some properties of $\log $. Now, we also know that rational number are the numbers which can be represented in the form of $\dfrac{p}{q}$, where $q \ne 0$ and irrational numbers are the numbers which cannot be represented in the form of $\dfrac{p}{q}$ . Now, we know that ${\log _{10}}{b^a} = a{\log _{10}}b$ and ${\log _{10}}{b^a} = x$ can be written as ${10^x} = {b^a}$. So, by using these properties, we will get our answer.

Complete step-by-step answer:
From the above question, we are given an expression that is $\log 100$ ,
Now, we will put $\log 100$ equals to $x$
$ \Rightarrow {\log _{10}}100 = x$
Now, simplifying the above equation,
We get,
$
   \Rightarrow {\log _{10}}100 = x \\
   \Rightarrow {\log _{10}}{10^2} = x \\
   \Rightarrow {10^x} = {10^2} \\
   \Rightarrow x = 2 \\
 $
Now, we get that $\log 100 = 2$
So, we know that 2 is a rational number, as it can be written as $\dfrac{2}{1}$ .

Hence, the expression given in the question is Rational.

Additional Information: In the above question, we have used the concepts of $\log $. Like ${\log _{10}}{b^a} = x$ can be written as ${10^x} = {b^a}$. We have also used the concepts of rational numbers and irrational numbers. As, we know that rational number are the numbers which can be represented in the form of $\dfrac{p}{q}$ , where $q \ne 0$ and irrational numbers are the numbers which cannot be represented in the form of $\dfrac{p}{q}$.

Note:
 In the above question, we have simplified the expression given in the question and then we have used the concepts of rational and irrational numbers. We have used properties of $\log $ in the question to simplify the expression. In the question where $\log $ Is given, always try to use the properties of it to simplify the expression to get a correct answer.