Question

${\log }_{2}7$ is a/an

• A. natural number
• B. integer
• C. irrational number
• D. rational number

Answer 1

The correct option is C 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 cannot be equal. So, our assumption was false.
${\log }_{2}7$ cannot be expressed in the form of $\frac{p}{q}$ where $p$ and $q$ are co prime integers.

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