x^y=z Denklemini Çözme

$$x ^ { y } = z$$

x için çözün (complex solution)

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

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

y için çözün (complex solution)

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

x için çözün

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

y için çözün

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