Discussion

Question:

The product of the roots of the equation $\log_{2} 2^{{(\log_{2} x)}^{2}}$ - 5log₂x + 6 = 0

Explanation:

Here, let log₂x = a

The given equation can be written as:
a² × log₂2 - 5a + 6 = 0
⇒ a² - 5a + 6 = 0 [log₂2 = 1]
⇒ a² - 3a - 2a + 6 = 0
⇒ a(a - 3) - 2(a - 3) = 0
⇒ (a - 3)(a - 2) = 0
⇒ a = 2 or 3

⇒ log₂x = 2 or log₂x = 3
⇒ x = 2² or x = 2³
⇒ x = 4 or x = 8

∴ Required product of the roots = 4 × 8 = 32

Hence, 32.

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