Question

Is $\log 2$ rational or irrational? Justify your answer.

Answer 1

Define rational number :

A rational number is a number, which can be written as $\frac{p}{q}$, where $p$and $q$ are integers , $q\ne 0$ and the H.C.F. of $p$ and $q$ is equal to $1$.

Let us assume $\log 2$ is ration number, that is,

$\log 2=\frac{p}{q}$………………………………………….equation$1$

where $p$and $q$are integers.

Since $\log 1=0$ and $\log 10=1$, $0<\log 2<1$

therefore, $p<q$

from equation$1$,

$$\begin{gathered} \Rightarrow 2={10}^{\frac{p}{q}} \\ \Rightarrow 2^q={(2\times 5)}^p \\ \Rightarrow 2^{q-p}=5^p \end{gathered}$$

where, $q-p$ is an integer greater than $0$.

Now, it can be seen that the L.H.S. is even and the R.H.S is odd.

Hence there is a contradiction and $\log 2$ is irrational.