Solve 3^-9x=9

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

Solve for x

$$x=-\frac{2}{9}\approx -0.222222222$$

Steps Using Definition of Logarithm

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

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

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(3).

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

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

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

Divide both sides by -9.

$$x=\frac{2}{-9}$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{9\ln(3)}-\frac{2}{9}$$

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