Question

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

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

Answer 1

The correct option is C an irrational number

Assume ${\log }_{2}7$ is a rational number.

Therefore, ${\log }_{2}7=\frac{a}{b}$ where, $a$ & $b$ are integers and $b\ne 0$

$\Rightarrow 2^a=7^b$

Since $2^a$ is even for all integers and $7^b$ is odd for all integers and no integer can be both even and odd.

Therefore, this is a contradiction and ${\log }_{2}7$ is an irrational number.