Solve 3^-1x-1=1/9

$$3 ^ { - 1 x - 1 } = \frac { 1 } { 9 }$$

Solve for x

$$x=1$$

Steps Using Definition of Logarithm

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

$$3^{-x-1}=\frac{1}{9}$$

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(3).

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

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

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

Add 1 to both sides of the equation.

$$-x=-2-\left(-1\right)$$

Divide both sides by -1.

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

Solve for x (complex solution)

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

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