NIMCET 2016 Question Paper with solutions

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Question : 52 of 120

Marks: +1, -0

The solution set of equation

\log_x

2

\log_{2x}

2 =

\log_{4x}

2is;

$2^{-√{2}}, 2^{√{2}}$
$\{1/2 , 2\}$
$\{1/4 , 2^2 \}$
$\{1/4 , 2\}$

Solution:

\log _{x} 2 \log _{2 x} 2=\log _{4 x} 2

∵ \log _{ {x}} 2

=\;{1}/{\log _{2} {x}}, \log _{2 {x}} 2

=\;{1}/{\log _{2} 2 {x}}

and

\log _{4 {x}} 2=\;{1}/{\log _{2} 4 {x}}

⇒ \;{1}/{\log _{2} {x}} * \;{1}/{\log _{2} 2 {x}}

=\;{1}/{\log _{2} 4 {x}}

⇒ \log _{2} {x} \log _{2} 2 {x}=\log _{2} 4 {x}

⇒ \log _{2} {x}(\log _{2} 2+\log _{2} {x})

=\log _{2} 4+\log _{2} {x}

Let

\log _{2} {x}

be

{a}

,

⇒ {a}(1+ {a})

=\log _{2} 2^{2}+ {a} \;\; ( ∵ \log _{2} 2=1)

⇒ {a}+ {a}^{2}=2+ {a}

⇒ a=± √{2}

⇒ \log _{2} {x}=± √{2}

⇒ x=2^{√{2}}, 2^{-√{2}}

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