y=n^k పరిష్కరించండి

$$y = n ^ { k }$$

kని పరిష్కరించండి (సంకీర్ణ పరిష్కారం)

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

nని పరిష్కరించండి (సంకీర్ణ పరిష్కారం)

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

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

kని పరిష్కరించండి

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

nని పరిష్కరించండి

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