y=x^m ઉકેલો

$$y = x ^ { m }$$

m માટે ઉકેલો (જટિલ સમાધાન)

$$\left\{\begin{matrix}m=\frac{2\pi n_{1}i}{\ln(x)}+\log_{x}\left(y\right)\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\m\in \mathrm{C}\text{, }&\left(x=0\text{ and }y=0\right)\text{ or }\left(x=1\text{ and }y=1\right)\end{matrix}\right.$$

x માટે ઉકેલો (જટિલ સમાધાન)

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

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

m માટે ઉકેલો

$$\left\{\begin{matrix}m=\log_{x}\left(y\right)\text{, }&y>0\text{ and }x\neq 1\text{ and }x>0\\m\in \mathrm{R}\text{, }&\left(x=1\text{ and }y=1\right)\text{ or }\left(x=-1\text{ and }y=-1\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=1\right)\\m>0\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

x માટે ઉકેલો

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