Solve 3^2x=1/9

$$3 ^ { 2 x } = \frac { 1 } { 9 }$$

Solve for x

$$x=-1$$

Steps Using Definition of Logarithm

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

$$3^{2x}=\frac{1}{9}$$

Take the logarithm of both sides of the equation.

$$\log(3^{2x})=\log(\frac{1}{9})$$

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

$$2x\log(3)=\log(\frac{1}{9})$$

Divide both sides by \log(3).

$$2x=\frac{\log(\frac{1}{9})}{\log(3)}$$

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

$$2x=\log_{3}\left(\frac{1}{9}\right)$$

Divide both sides by 2.

$$x=-\frac{2}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(3)}-1$$

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