Prove that number `(log)_2 7` is an irrational number.

Options

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

Answer

C

Explanation

Let `"log"_(2)7` be a rational number equal to `(m)/(n)`, where `m, n notin N` and they do not have a common factor. Then,
`"log"_(2) 7 = (m)/(n) rArr 7 = 2^(m//n) rArr 7^(n) = 2^(m)`
Clearly, this is impossible, because `7^(th)` is an natural number and `2^(nd)` is an even natural number.
Hence, `"log"_(2)7` is an irrational number.