The number $${\log _2}$$ $$7$$ is

Solution :
We know that $${\log _2}$$ $$4 = 2$$  and $${\log _2} 8 = 3$$
∴ $${\log _2}$$ $$7$$ lies between 2 and 3
∴ $${\log _2}$$ $$7$$ is either rational or irrational but not integer or prime number
If possible let $${\log _2}$$ $$7 = \frac{p}{q}$$  (a rational number)
$$\eqalign{ & \Rightarrow \,{2^{\frac{p}{q}}} = 7 \cr & \Rightarrow \,{2^p} = {7^q} \cr} $$
$$ \Rightarrow $$ even number = odd number
∴ We get a contradiction, so assumption is wrong.
Hence $${\log _2}$$ $$7$$ must be an irrational number.