Question

The number ${\log }_{2}7$ is _____

• A. a natural number
• B. a rational number
• C. an irrational number
• D. a negative integer

Answer 1

The correct option is C

an irrational number

Suppose ${\log }_{2}7$ is a rational number; say $\frac{p}{q}$ where p and q are integer prime to each other.

Then, $\frac{p}{q}$= ${\log }_{2}7$

$7=2^{p/q}$

$7^q$ = $2^p$

The integral powers of an odd number are all odd $(7^2=49,7^3=343)$ and the integral powers of an even number are all even $(4^2=16,4^3=64)$.

Since RHS is even number and LHS is odd. Both are not equal. So, our assumption was false. ${\log }_{2}7$ cannot be expressed in the form of $\frac{p}{q}$ where $p$ and $q$ are integers or prime.

So,${\log }_{2}7$ is an irrational number.