解決b^x=c |Microsoft數學求解器

$$b ^ { x } = c$$

解 b (復數求解)

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

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

解 c (復數求解)

$$c=b^{x}$$

解 b

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

解 c

$$c=b^{x}$$

$$\left(b<0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(b=0\text{ and }x>0\right)\text{ or }b>0$$