Solve 0=1+x_{H}+z^xH

$$0 = 1 + x _ { H } + z ^ { x } H$$

Solve for H (complex solution)

$$\left\{\begin{matrix}H=-\frac{x_{H}+1}{z^{x}}\text{, }&x=0\text{ or }z\neq 0\\H\in \mathrm{C}\text{, }&x_{H}=-1\text{ and }z=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$1+x_{H}+z^{x}H=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$x_{H}+z^{x}H=-1$$

Subtract x_{H} from both sides.

$$z^{x}H=-1-x_{H}$$

The equation is in standard form.

$$z^{x}H=-x_{H}-1$$

Divide both sides by z^{x}.

$$\frac{z^{x}H}{z^{x}}=\frac{-x_{H}-1}{z^{x}}$$

Dividing by z^{x} undoes the multiplication by z^{x}.

$$H=\frac{-x_{H}-1}{z^{x}}$$

Divide -1-x_{H} by z^{x}.

$$H=-\frac{x_{H}+1}{z^{x}}$$

Solve for H

$$\left\{\begin{matrix}H=-\frac{x_{H}+1}{z^{x}}\text{, }&z>0\text{ or }\left(Denominator(x)\text{bmod}2=1\text{ and }z<0\right)\\H\in \mathrm{R}\text{, }&x_{H}=-1\text{ and }z=0\text{ and }x>0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$1+x_{H}+z^{x}H=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$x_{H}+z^{x}H=-1$$

Subtract x_{H} from both sides.

$$z^{x}H=-1-x_{H}$$

The equation is in standard form.

$$z^{x}H=-x_{H}-1$$

Divide both sides by z^{x}.

$$\frac{z^{x}H}{z^{x}}=\frac{-x_{H}-1}{z^{x}}$$

Dividing by z^{x} undoes the multiplication by z^{x}.

$$H=\frac{-x_{H}-1}{z^{x}}$$

Divide -1-x_{H} by z^{x}.

$$H=-\frac{x_{H}+1}{z^{x}}$$

Solve for x (complex solution)

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

Solve for x

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