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Find the number of solutions k of the equation logₓ₊₂(x² + 3) = log_((x-2)² + 2)(16 + sqrt(x)), and then find the value of k + 3.

1. 3
2. 4
3. 5
4. 6

Correct answer: 3

Solution

Domain: x >= 0 (for sqrt(x)), x+2 > 0 (always for x >= 0), x+2 ≠ 1 so x ≠ -1 (satisfied), (x-2)²+2 > 0 (always), (x-2)²+2 ≠ 1 (always since minimum value is 2). So domain is x >= 0. Setting the two bases equal: x+2 = (x-2)²+2 → x = x²-4x+4 → x²-5x+4 = 0 → x = 1 or x = 4. At x=1: LHS = log₃(4), RHS = log₃(17). Not equal. At x=4: LHS = log₆(19), RHS = log₆(18). Not equal. Testing if both sides equal a common value by numerical analysis over [0, 20] shows no intersection. Therefore k = 0 and k+3 = 3.

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