解決4^x-2^x=y |Microsoft數學求解器

$$4 ^ { x } - 2 ^ { x } = y$$

解 x (復數求解)

$$\left\{\begin{matrix}\\x=\log_{2}\left(\frac{\sqrt{4y+1}+1}{2}\right)+\frac{2\pi n_{2}i}{\ln(2)}\text{, }n_{2}\in \mathrm{Z}\text{, }&\text{unconditionally}\\x=\log_{2}\left(\frac{-\sqrt{4y+1}+1}{2}\right)+\frac{2\pi n_{1}i}{\ln(2)}\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\end{matrix}\right.$$

解 y (復數求解)

$$y=4^{x}-2^{x}$$

解 x

$$\left\{\begin{matrix}x=\log_{2}\left(\frac{\sqrt{4y+1}+1}{2}\right)\text{, }&y\geq -\frac{1}{4}\\x=\log_{2}\left(\frac{-\sqrt{4y+1}+1}{2}\right)\text{, }&y\geq -\frac{1}{4}\text{ and }y<0\end{matrix}\right.$$

解 y

$$y=2^{x}\left(2^{x}-1\right)$$