A Nice Algebra Problem | Math Olympiad | A Nice Exponential Equation

[@aminimam5118]

4^(-1)= (1/2)^2=x^(1/x) x=1/2

[@SALogics]

Very nice! ❤

[@zhaoyue5953]

How to solve 4^-x = x (4^-x)^1/-x = x^1/-x => 4 = x^1/-x (1) let: 1/x = a, so -a = -1/x, x=1/a. (2) put (2) in (1) => 4 = (1/a)^-a, 4 = (1/a)^-1xa, 4 = [(1/a)^-1]^a, note: (1/a)^-1 = a so: 4 = [(1/a)^-1]^a = a^a 2^2=a^a => a=2 put a value 2 in (2): 1/x = a => 1/x = 2 => x = 1/2 x = 1/2 is the answer

[@SALogics]

Very nice! ❤

[@RossJandorf]

4 to power of-x =1/4 to power of x, x equals 1/2

[@SALogics]

Very Nice! ❤

[@walterwen2975]

A Nice Exponential Equation: 4^(- x) = x; x =? [4^(- x)]^(- 1/x) = x^(- 1/x), x^(- 1/x) = 4 = 2^2 = 2^[(- 1)(- 2)] = [2^(- 1)]^(- 2) x^(- 1/x) = 4 = [2^(- 1)]^(- 2) = (1/2)^[- 1/(1/2)]; x = 1/2 Answer check: x = 1/2: 4^(- x) = 4^(- 1/2) = 2^(- 1) = 1/2 = x; Confirmed Final answer: x = 1/2

[@SALogics]

Very nice! ❤

[@trojanleo123]

x = 1/2

[@SALogics]

Yes, you are right! ❤