A Fascinating Exponential Equation- Olympiad Algebra Challenge

[@tijanimaths6006]

ممتاز excellent. Great

[@sunil.shegaonkar1]

It is interesting to see two roots at 2:41

[@tejpalsingh366]

x= + -2; + - √2

[@user-ny6jf9is3t]

χ=-2, 2, ριζα2, -ριζα2

[@SidneiMV]

16^x² = x¹⁶ = (x²)⁸ x² = u 16ᵘ= u⁸ 2⁴ᵘ= (u²)⁴ 2ᵘ= u² uln2 = 2lnu (1/u)lnu = (1/2)ln2 (1/u)ln(1/u) = (1/2)ln(1/2) = (1/4)ln(1/4) => u = 2 or u = 4 u = 2 => x = ± √2 u = 4 => x = ± 2

[@user-kp2rd5qv8g]

Let t=x^2. Then, t ln16 = 8 ln t > 1/t ln t = 1/2 ln 2 > 1/t ln (1/t) = 1/2 ln (1/2) > ln (1/t) = W(1/2 ln(1/2)), where W is the Lambert W function. So t=e^[-W(1/2 ln(1/2))] > x = e^[-1/2 W(1/2 ln(1/2))] = √2, +/-2. So, x=√2, +/-2.

[@RajeshKumar-wu7ox]

X=2

[@RealQinnMalloryu4]

X^16 x^4^4 x^2^2^2^2 x^1^2^1^1 x^2^1 (x ➖ 2x+1).