Solve 7^x+3y-5z=343

$${ 7 }^{ x+3y-5z } =343$$

Solve for x

$$x=3+5z-3y$$

Steps Using Definition of Logarithm

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

$$7^{x+3y-5z}=343$$

Take the logarithm of both sides of the equation.

$$\log(7^{x+3y-5z})=\log(343)$$

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

$$\left(x+3y-5z\right)\log(7)=\log(343)$$

Divide both sides by \log(7).

$$x+3y-5z=\frac{\log(343)}{\log(7)}$$

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

$$x+3y-5z=\log_{7}\left(343\right)$$

Subtract 3y-5z from both sides of the equation.

$$x=3-\left(3y-5z\right)$$

Solve for y

$$y=\frac{5z}{3}-\frac{x}{3}+1$$

Steps Using Definition of Logarithm

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

$$7^{3y+x-5z}=343$$

Take the logarithm of both sides of the equation.

$$\log(7^{3y+x-5z})=\log(343)$$

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

$$\left(3y+x-5z\right)\log(7)=\log(343)$$

Divide both sides by \log(7).

$$3y+x-5z=\frac{\log(343)}{\log(7)}$$

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

$$3y+x-5z=\log_{7}\left(343\right)$$

Subtract x-5z from both sides of the equation.

$$3y=3-\left(x-5z\right)$$

Divide both sides by 3.

$$y=\frac{3+5z-x}{3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(7)}+5z-3y+3$$

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

Solve for y (complex solution)

$$y=\frac{2\pi n_{1}i}{3\ln(7)}+\frac{5z}{3}-\frac{x}{3}+1$$

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