If x and y are positive real numbers such that logx (x2 + 12) = 4 and 3logy x = 1, then x + y equals?
Explanation:
Given, logx (x2 + 12) = 4 ⇒ x2 + 12 = x4 ⇒ a + 12 = a2 [Take x2 = a] ⇒ a2 – a – 12 = 0 ⇒ (a – 4)(a + 3) = 0 ⇒ a = -3 or 4. [a = x2 cannot be negative hence -3 is rejected] ⇒ x2 = 4 ⇒ x = ± 2 [x = -2 is rejected as log is not defined for negative numbers] ⇒ x = 2
Also, 3logy x = 1 ⇒ 3logy 2 = 1 ⇒ logy 23 = 1 ⇒ 23 = y1 ⇒ y = 8
∴ x + y = 2 + 8 = 10
Hence, option (d).
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