Japan | A Nice Algebra Problem | Math Olympiad 

how come, 7^x = 11^x ???

X=0.

A Nice Algebra Problem: 7ˣ - 11ˣ = √(77ˣ - 121ˣ), x = ? Let: u = 7ˣ, v = 11ˣ; 7ˣ - 11ˣ = √(77ˣ - 121ˣ) = √(11ˣ)√(7ˣ - 11ˣ) = √[v(u - v)] (u - v)² = u² - 2uv + v² = v(u - v) = uv - v², u² - 3uv + 2v² = 0 (u² - uv) - (2uv - 2v²) = u(u - v) - 2v(u - v) = (u - v)(u - 2v) = 0 u - v = 0, u = v; 7ˣ = 11ˣ or u - 2v = 0, u = 2v; 7ˣ = 2(11ˣ) (11ˣ)/(7ˣ) = (11/7)ˣ = 1 = (11/7)⁰ or (11/7)ˣ = 1/2 x = 0 or x = log(1/2)/log(11/7) = - 0.301/0.196 = - 1.534 Answer check: x = 0: 7ˣ - 11ˣ = 7⁰ - 11⁰ = 1 - 1 = 0, √(77ˣ - 121ˣ) = √(1 - 1) = 0; Confirmed x = - 1.534: 7^(- 1.534) - 11^(- 1.534) = 0.051 - 0.025 = 0.026 √[77^(- 1.534) - 121^(- 1.534)] = √(0.0013 - 0.0006) = 0.025; Confirmed The calculation was achieved on a smartphone with a standard calculator app Final answer: x = 0 or x = - 1.534