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Byju's Answer
Standard XI
Mathematics
Logarithmic Inequalities
If log x-2 6+...
Question
If
log
x
−
2
6
+
log
x
+
2
6
>
log
x
−
2
6
⋅
log
x
+
2
6
, 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,
x
−
2
>
0
,
x
+
2
>
0
,
x
−
2
≠
1
,
x
+
2
≠
1
⇒
x
>
2
,
x
>
−
2
,
x
≠
3
,
x
≠
−
1
⇒
x
∈
(
2
,
∞
)
−
{
3
}
⋯
(
1
)
Now,
log
x
−
2
6
+
log
x
+
2
6
>
log
x
−
2
6
⋅
log
x
+
2
6
⇒
1
log
6
(
x
−
2
)
+
1
log
6
(
x
+
2
)
>
1
log
6
(
x
−
2
)
log
6
(
x
+
2
)
⇒
log
6
(
x
+
2
)
(
x
−
2
)
log
6
(
x
−
2
)
log
6
(
x
+
2
)
−
1
log
6
(
x
−
2
)
log
6
(
x
+
2
)
>
0
⇒
log
6
(
x
2
−
4
)
−
1
log
6
(
x
−
2
)
log
6
(
x
+
2
)
>
0
Consider
log
6
(
x
2
−
4
)
−
1
=
0
⇒
x
2
−
4
=
6
⇒
x
=
√
10
[
From
(
1
)
]
log
6
(
x
−
2
)
=
0
⇒
x
−
2
=
1
⇒
x
=
3
log
6
(
x
+
2
)
=
0
⇒
x
+
2
=
1
⇒
x
=
−
1
Not possible
∴
x
∈
(
−
∞
,
3
)
∪
(
√
10
,
∞
)
⋯
(
2
)
From
(
1
)
and
(
2
)
,
x
∈
(
2
,
3
)
∪
(
√
10
,
∞
)
Suggest Corrections
0
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Q.
If
log
x
−
2
6
+
log
x
+
2
6
>
log
x
−
2
6
⋅
log
x
+
2
6
, then
x
∈
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o
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.
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∞
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