Solve 2z=ze^xy

$$2z=z { e }^{ xy }$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\ln(2)}{y}\text{, }&y\neq 0\\x\in \mathrm{R}\text{, }&z=0\end{matrix}\right.$$

Steps Using Definition of Logarithm

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

$$ze^{xy}=2z$$

Use the rules of exponents and logarithms to solve the equation.

$$ze^{yx}=2z$$

Divide both sides by z.

$$e^{yx}=2$$

Take the logarithm of both sides of the equation.

$$\log(e^{yx})=\log(2)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$yx\log(e)=\log(2)$$

Divide both sides by \log(e).

$$yx=\frac{\log(2)}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$yx=\log_{e}\left(2\right)$$

Divide both sides by y.

$$x=\frac{\ln(2)}{y}$$

Solve for y

$$\left\{\begin{matrix}y=\frac{\ln(2)}{x}\text{, }&x\neq 0\\y\in \mathrm{R}\text{, }&z=0\end{matrix}\right.$$

Steps Using Definition of Logarithm

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

$$ze^{xy}=2z$$

Divide both sides by z.

$$e^{xy}=2$$

Take the logarithm of both sides of the equation.

$$\log(e^{xy})=\log(2)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$xy\log(e)=\log(2)$$

Divide both sides by \log(e).

$$xy=\frac{\log(2)}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$xy=\log_{e}\left(2\right)$$

Divide both sides by x.

$$y=\frac{\ln(2)}{x}$$