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Question:

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

Explanation:

u = (log₂x)² – 6log₂x + 12   ...(1)

x^u = 256

Taking log on both sides,

ulog₂x = log₂256 = 8

Let log₂x = m

∴ u × m = 8

Substituting in (1),

$\frac{8}{m}$ = 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).