Question

The number $lo{g}_{2}7$ 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

Suppose, if possible, $lo{g}_{2}7$ is rational, say $\frac{p}{q}$ where p and q are integers, prime to each other.
$Then,\frac{p}{q}=lo{g}_{2}7\Rightarrow 7=2^{\frac{p}{q}}\Rightarrow 2^p=7^q$
which is false since L.H.S is even and R.H.S is odd.
Obviously $lo{g}_{2}7$ is not an integer and hence not a prime number.