If the sum of all solutions of the equation ₂( x) + ₂( x) = -1 in the interval [0, 8 ] is k , then the value of k is _______.

Step-by-step solution

For the given logarithmic equation to be defined, the arguments must be strictly positive: x > 0 and x > 0 This restricts the valid solutions to the first quadrant. Using the properties of logarithms, we can combine the terms: ₂( x x) = -1 x x = 2⁻¹ = 1 2 Multiplying both sides by 2: 2 x x = 1 (2x) = 1 The general solution for this is: 2x = 2 + 2n x = 4 + n , where n Z Since x must lie in the first quadrant, n must be an even integer. Let n = 2m , so x = 4 + 2m . We need to find solutions in the interval [0, 8 ] :