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Question

If logx26+logx+26>logx26logx+26, then x

A
(,3)
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B
(1,3)
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C
(10,)
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D
(2,3)
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Solution

The correct options are
C (10,)
D (2,3)
For the inequality to be defined,
x2>0, x+2>0, x21, x+21
x>2, x>2, x3, x1
x(2,){3} (1)

Now, logx26+logx+26>logx26logx+26
1log6(x2)+1log6(x+2)>1log6(x2)log6(x+2)

log6(x+2)(x2)log6(x2)log6(x+2)1log6(x2)log6(x+2)>0

log6(x24)1log6(x2)log6(x+2)>0

Consider log6(x24)1=0
x24=6
x=10 [ From(1) ]

log6(x2)=0
x2=1x=3

log6(x+2)=0
x+2=1x=1 Not possible


x(,3)(10,) (2)
From (1) and (2),
x(2,3)(10,)

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