Solve 3^m+2n=2

$$3 ^ { m + 2 n } = 2$$

Solve for m

$$m=-2n+\log_{3}\left(2\right)$$

Steps Using Definition of Logarithm

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

$$3^{m+2n}=2$$

Take the logarithm of both sides of the equation.

$$\log(3^{m+2n})=\log(2)$$

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

$$\left(m+2n\right)\log(3)=\log(2)$$

Divide both sides by \log(3).

$$m+2n=\frac{\log(2)}{\log(3)}$$

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

$$m+2n=\log_{3}\left(2\right)$$

Subtract 2n from both sides of the equation.

$$m=\log_{3}\left(2\right)-2n$$

Solve for n

$$n=\frac{-m+\log_{3}\left(2\right)}{2}$$

Steps Using Definition of Logarithm

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

$$3^{2n+m}=2$$

Take the logarithm of both sides of the equation.

$$\log(3^{2n+m})=\log(2)$$

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

$$\left(2n+m\right)\log(3)=\log(2)$$

Divide both sides by \log(3).

$$2n+m=\frac{\log(2)}{\log(3)}$$

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

$$2n+m=\log_{3}\left(2\right)$$

Subtract m from both sides of the equation.

$$2n=\log_{3}\left(2\right)-m$$

Divide both sides by 2.

$$n=\frac{-m+\log_{3}\left(2\right)}{2}$$