Solve 3^-2x=1/81

$$3 ^ { - 2 x } = \frac { 1 } { 81 }$$

Solve for x

$$x=2$$

Steps Using Definition of Logarithm

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

$$3^{-2x}=\frac{1}{81}$$

Take the logarithm of both sides of the equation.

$$\log(3^{-2x})=\log(\frac{1}{81})$$

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

$$-2x\log(3)=\log(\frac{1}{81})$$

Divide both sides by \log(3).

$$-2x=\frac{\log(\frac{1}{81})}{\log(3)}$$

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

$$-2x=\log_{3}\left(\frac{1}{81}\right)$$

Divide both sides by -2.

$$x=-\frac{4}{-2}$$

Solve for x (complex solution)

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

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