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Question
If √log2x−0.5=log2√x, then x is
If √log2x−0.5=log2√x, then x is
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Solution
The correct option is B a prime number
Clearly, x>0, √x>0
⇒x>0 ⋯(1)
and log2x≥0
⇒x≥1 ⋯(2)
So, from (1) and (2), x≥1
√log2x−0.5=log2√x
⇒√log2x=0.5+0.5log2x
⇒log2x=14(1+2log2x+(log2x)2)
⇒4log2x=1+2log2x+(log2x)2
⇒(log2x−1)2=0
⇒log2x=1
⇒x=2
Clearly, x>0, √x>0
⇒x>0 ⋯(1)
and log2x≥0
⇒x≥1 ⋯(2)
So, from (1) and (2), x≥1
√log2x−0.5=log2√x
⇒√log2x=0.5+0.5log2x
⇒log2x=14(1+2log2x+(log2x)2)
⇒4log2x=1+2log2x+(log2x)2
⇒(log2x−1)2=0
⇒log2x=1
⇒x=2
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