Solve 3^-x=0.2

$$3 ^ { - x } = 0.2$$

Solve for x

$$x=\log_{3}\left(5\right)\approx 1.464973521$$

Steps Using Definition of Logarithm

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

$$3^{-x}=0.2$$

Take the logarithm of both sides of the equation.

$$\log(3^{-x})=\log(0.2)$$

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

$$-x\log(3)=\log(0.2)$$

Divide both sides by \log(3).

$$-x=\frac{\log(0.2)}{\log(3)}$$

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

$$-x=\log_{3}\left(0.2\right)$$

Divide both sides by -1.

$$x=-\frac{\log_{3}\left(5\right)}{-1}$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{\ln(3)}+\log_{3}\left(5\right)$$

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