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Determine the number of solutions of the equation log₂(x² + 3) = (1/2)*log_(1/3)(x + 1/x) for x > 0.

1. 0
2. 1
3. 2
4. infinite

Correct answer: 0

Solution

For x>0: LHS = log₂(x²+3) >= log₂(3) > 1 > 0. RHS = (1/2)*log_(1/3)(x+1/x) = -(1/2)*log₃(x+1/x). By AM-GM, x+1/x >= 2 > 1, so log₃(x+1/x) > 0, making RHS < 0. Since LHS > 0 and RHS < 0 for all x > 0, the equation has no solution.

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