Solve 9^x-5=1/81

$$9 ^ { x - 5 } = \frac { 1 } { 81 }$$

Solve for x

$$x=3$$

Steps Using Definition of Logarithm

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

$$9^{x-5}=\frac{1}{81}$$

Take the logarithm of both sides of the equation.

$$\log(9^{x-5})=\log(\frac{1}{81})$$

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

$$\left(x-5\right)\log(9)=\log(\frac{1}{81})$$

Divide both sides by \log(9).

$$x-5=\frac{\log(\frac{1}{81})}{\log(9)}$$

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

$$x-5=\log_{9}\left(\frac{1}{81}\right)$$

Add 5 to both sides of the equation.

$$x=-2-\left(-5\right)$$

Solve for x (complex solution)

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

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