Solve 9(3^-x+1)=81

$$9 ( 3 ^ { - x + 1 } ) = 81$$

Solve for x

$$x=-1$$

Steps Using Definition of Logarithm

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

$$9\times 3^{-x+1}=81$$

Divide both sides by 9.

$$3^{-x+1}=9$$

Take the logarithm of both sides of the equation.

$$\log(3^{-x+1})=\log(9)$$

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

$$\left(-x+1\right)\log(3)=\log(9)$$

Divide both sides by \log(3).

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

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

$$-x+1=\log_{3}\left(9\right)$$

Subtract 1 from both sides of the equation.

$$-x=2-1$$

Divide both sides by -1.

$$x=\frac{1}{-1}$$

Solve for x (complex solution)

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

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