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Byju's Answer
Standard XII
Mathematics
Logarithmic Inequalities
If √log2x-0.5...
Question
If
√
log
2
x
−
0.5
=
log
2
√
x
,
then
x
is
A
an odd integer
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B
a prime number
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C
a composite number
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D
an irrational number
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Solution
The correct option is
B
a prime number
Clearly,
x
>
0
,
√
x
>
0
⇒
x
>
0
⋯
(
1
)
and
log
2
x
≥
0
⇒
x
≥
1
⋯
(
2
)
So, from
(
1
)
and
(
2
)
,
x
≥
1
√
log
2
x
−
0.5
=
log
2
√
x
⇒
√
log
2
x
=
0.5
+
0.5
log
2
x
⇒
log
2
x
=
1
4
(
1
+
2
log
2
x
+
(
log
2
x
)
2
)
⇒
4
log
2
x
=
1
+
2
log
2
x
+
(
log
2
x
)
2
⇒
(
log
2
x
−
1
)
2
=
0
⇒
log
2
x
=
1
⇒
x
=
2
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Q.
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√
log
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x
−
0.5
=
log
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√
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