Japan | A Nice Algebra Problem | Math Olympiad

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[@AlamKhan-xc4fh]

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

[@walterwen2975]

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

[@MD-kv9zo]

X=0.