For x ≥ y > 1, let log_x(x/y) + log_y(y/x) = k, then the value of k can never be equal to

Let log_x y = t. Since x ≥ y > 1, we have 0 < t ≤ 1. Then log_x(x/y) = 1 – t and log_y(y/x) = 1 – log_y x = 1 – 1/t. So k = (1–t) + (1 – 1/t) = 2 – t – 1/t. By AM-GM, t + 1/t ≥ 2 with equality when t = 1, so k ≤ 0. The maximum value of k is 0 (when t = 1, i.e., x = y). For t ∈ (0, 1], k ranges from –∞ to 0. Therefore k can equal –1, –1/2, 0 but never 1.