The equation x(3/4)(log2 x)2+(log2 x)-5/4 =2 has

For the given equation to be meaningful we must have x>0. For x>0 the given equation can be written as 34(log2 x)2+log2 x-54=logx 2=12logx 2⇒34t2+t-54=12(1t)By putting t=log2 x so that logx 2=1t becauselog2 x logx2=1. ⇒3t3+4t2-5t-2=0⇒(t-1)(t+2)(3t+1)=0 ⇒log2 x=t=1, -2, - 13 ⇒x=2, 2-2, 2-1/3 or x=2, 14, 121/3Thus the given equation has exactly three real solutions out of which exactly one is irrational namely 121/3.