The number _2 7 is - Vidyadip

MCQ

The number ${\log _2}7$ is

A
An integer
B
A rational number
✓
An irrational number
D
A prime number

✓

Answer

Correct option: C.

An irrational number

c
(c) Suppose, if possible, ${\log _2}7$ is rational, say $p/q$ where $p$ and $q$ are integers, prime to each other.

Then, ${p \over q} = {\log _2}7\,\,\,\,\, \Rightarrow 7 = {2^{p/q}}\,\,\, \Rightarrow {2^p} = {7^q}$,

which is false since $L.H.S$ is even and $R.H.S$ is odd. Obviously ${\log _2}7$ is not an integer and hence not a prime number.

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