حل f^x=2x^2-4x+1

$${ f }^{ x } = 2 { x }^{ 2 } -4x+1$$

حل مسائل f (complex solution)

$$f=e^{\frac{Im(x)arg(2x^{2}-4x+1)+iRe(x)arg(2x^{2}-4x+1)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}iRe(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}Im(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\left(|2x^{2}-4x+1|\right)^{\frac{Re(x)-iIm(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}$$

$$n_{1}\in \mathrm{Z}$$

حل مسائل f

$$\left\{\begin{matrix}f=\left(2x^{2}-4x+1\right)^{\frac{1}{x}}\text{, }&\left(\left(2x^{2}-4x+1\right)^{\frac{1}{x}}>0\text{ and }x>\frac{\sqrt{2}}{2}+1\right)\text{ or }\left(\left(2x^{2}-4x+1\right)^{\frac{1}{x}}>0\text{ and }x\neq 0\text{ and }x<-\frac{\sqrt{2}}{2}+1\right)\text{ or }\left(\left(2x^{2}-4x+1\right)^{\frac{1}{x}}>0\text{ and }x\leq -\frac{\sqrt{2}}{2}+1\text{ and }x>0\right)\text{ or }x=\frac{\sqrt{2}}{2}+1\text{ or }x=-\frac{\sqrt{2}}{2}+1\text{ or }\left(\left(2x^{2}-4x+1\right)^{\frac{1}{x}}<0\text{ and }x>\frac{\sqrt{2}}{2}+1\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(x\neq 0\text{ and }\left(2x^{2}-4x+1\right)^{\frac{1}{x}}<0\text{ and }x<-\frac{\sqrt{2}}{2}+1\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(\left(2x^{2}-4x+1\right)^{\frac{1}{x}}<0\text{ and }x\leq -\frac{\sqrt{2}}{2}+1\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x>0\right)\text{ or }\left(Numerator(x)\text{bmod}2=1\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x<\frac{\sqrt{2}}{2}+1\text{ and }x>-\frac{\sqrt{2}}{2}+1\right)\\f=-\left(2x^{2}-4x+1\right)^{\frac{1}{x}}\text{, }&\left(Numerator(x)\text{bmod}2=1\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x<\frac{\sqrt{2}}{2}+1\text{ and }x>-\frac{\sqrt{2}}{2}+1\right)\text{ or }\left(\left(2x^{2}-4x+1\right)^{\frac{1}{x}}>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x>\frac{\sqrt{2}}{2}+1\right)\text{ or }\left(x\neq 0\text{ and }\left(2x^{2}-4x+1\right)^{\frac{1}{x}}>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x<-\frac{\sqrt{2}}{2}+1\right)\text{ or }\left(x\leq -\frac{\sqrt{2}}{2}+1\text{ and }\left(2x^{2}-4x+1\right)^{\frac{1}{x}}>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x>0\right)\text{ or }\left(x\leq -\frac{\sqrt{2}}{2}+1\text{ and }\left(2x^{2}-4x+1\right)^{\frac{1}{x}}<0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }x>0\right)\text{ or }\left(x\neq 0\text{ and }\left(2x^{2}-4x+1\right)^{\frac{1}{x}}<0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }x<-\frac{\sqrt{2}}{2}+1\right)\text{ or }\left(\left(2x^{2}-4x+1\right)^{\frac{1}{x}}<0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }x>\frac{\sqrt{2}}{2}+1\right)\\f\neq 0\text{, }&x=0\end{matrix}\right.$$