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Question

The equation 4logx2(x)+2log4x(x2)=3log2x(x3) is having 
  1. all rational roots
  2. two rational and one irrational roots
  3. one rational and two irrational roots
  4. all irrational roots


Solution

The correct option is A all rational roots
4logx2(x)+2log4x(x2)=3log2x(x3)
4log2(x)log2(x/2)+2log2(x2)log2(4x)=3log2(x3)log2(2x)
4×12log2(x)log2(x)1+4log2(x)2+log2(x)=9log2(x)1+log2(x)

Put log2x=t
2tt1+4tt+2=9tt+1
t=0  or  2t1+4t+2=9t+1
2t+4+4t4(t1)(t+2)=9t+1
t2+t6=0
(t2)(t+3)=0
t=0,2,3
x=1,4,18

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