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CAT 2004 — QA Question 31

LogarithmsEasy

Let u = (log2 x)2 – 6 log2 x + 12 where x is a real number. Then the equation xu = 256, has

Answer & solution

Correct answer: exactly one solution for x

  • A

    no solution for x

  • exactly one solution for x

  • C

    exactly two distinct solutions for x

  • D

    exactly three distinct solutions for x

Solution

u = (log2x)2 – 6log2x + 12   ...(1)

xu = 256

Taking log on both sides,

ulog2x = log2256 = 8

Let log2x = m

∴ u × m = 8

Substituting in (1),

8m = m2 - 6m + 12

⇒ m3 – 6m2 + 12m – 8 = 0
⇒ (m – 2)3 = 0
⇒ m = 2
⇒ log2x = 2
⇒ x = 4

∴ The given equation has exactly one solution for x.

Hence, option (b).

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CAT 2004 QA Q31: Let u = (log 2 x ) 2 – 6 log 2 x + 12 where x is a real number. Then the equation x u = 256, has — Solution | TheCATExam