CAT Exam Model Paper 1 with solutions for free online practice

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Question : 97 of 100

Marks: +1, -0

What is the number of solutions for real x, which satisfy the equation

2 \log _ { 2 } \log _ { 2 } x + \log _ { 1 / 2 } \log _ { 2 } ( 2 √ { 2 x } ) = 1 ?

Your Answer:

Solution:

2 \log _ { 2 } \log _ { 2 } x + \log _ { 1 / 2 } \log _ { 2 } ( 2 √ { 2 x } ) = 1

⇒

2 \log _ { 2 } \log _ { 2 } x - \log _ { 2 } \log _ { 2 } ( 2 √ { 2 } x ) = \log _ { 2 } 2

⇒

\log _ { 2 } ( \log _ { 2 } x ) ^ { 2 } - \log _ { 2 } [ \log _ { 2 } 2 √ { 2 } x ] = \log _ { 2 } 2

⇒

\log _ { 2 } { ( \log _ { 2 } x ) ^ { 2 } }/ { [ \log _ { 2 } 2√ { 2 } x ] } = \log _ { 2 } 2

⇒

{ ( \log _ { 2 } x ) ^ { 2 } }/ { \log _ { 2 } ( 2√{ 2 } x ) } = 2

⇒

( \log _ { 2 } x ) ^ { 2 } = 2 \log _ { 2 } ( 2 √{ 2 } x ) = 2 \log _ { 2 } ( 2 ^ { 3 / 2 } x )

⇒

( \log _ { 2 } x ) ^ { 2 } = 2 [ 3 / 2 \log _ { 2 } 2 + \log _ { 2 } x ] = 3 + 2 \log _ { 2 } x

⇒ ( \log _ { 2 } x ) ^ { 2 } - 2 \log _ { 2 } x - 3 = 0

\log _ { 2 } x = - 1

or

\log _ { 2 } x = 3

⇒

x = { 1 }/ { 2 }

or

x = 8

But for x =

1/2

, log

_2

log

_2

1/2

is undefined
∴ Only possible value of x = 8.

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