Solve 3^-b=3^2

$$3 ^ { - b } = 3 ^ { 2 }$$

Solve for b

$$b=-2$$

Steps Using Definition of Logarithm

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

$$3^{-b}=9$$

Take the logarithm of both sides of the equation.

$$\log(3^{-b})=\log(9)$$

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

$$-b\log(3)=\log(9)$$

Divide both sides by \log(3).

$$-b=\frac{\log(9)}{\log(3)}$$

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

$$-b=\log_{3}\left(9\right)$$

Divide both sides by -1.

$$b=\frac{2}{-1}$$

Solve for b (complex solution)

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

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