Rešite 1-m^k=x

$$1 - m ^ { k } = x$$

Rešitev za k (complex solution)

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

Rešitev za m (complex solution)

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

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

Rešitev za k

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

Rešitev za m

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