The number of distinct solutions of the equation 2 _ x ( x) + _ x ( x) = 3 in the interval x [0, 8 ] is ______.

Step-by-step solution

The given equation is 2 _ x ( x) + _ x ( x) = 3 . For the logarithms to be defined, we must have the base and argument strictly positive and the base not equal to 1 . Thus, the domain constraints are: x > 0 , x 1 x > 0 , x 1 These conditions imply that x must strictly lie in the first quadrant, i.e., x (2n , 2n + 2 ) for integer n . Let t = _ x ( x) . Then, by the base-change property, _ x ( x) = 1 t . The equation becomes: 2t + 1 t = 3 2t^2 - 3t + 1 = 0 (2t - 1)(t - 1) = 0 This gives t = 1 or t = 1 2 . Case 1: t =