Solve 81^x-2=27

$$81 ^ { x - 2 } = 27$$

Solve for x

$$x = \frac{11}{4} = 2\frac{3}{4} = 2.75$$

Steps Using Definition of Logarithm

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

$$81^{x-2}=27$$

Take the logarithm of both sides of the equation.

$$\log(81^{x-2})=\log(27)$$

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

$$\left(x-2\right)\log(81)=\log(27)$$

Divide both sides by \log(81).

$$x-2=\frac{\log(27)}{\log(81)}$$

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

$$x-2=\log_{81}\left(27\right)$$

Add 2 to both sides of the equation.

$$x=\frac{3}{4}-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2\ln(3)}+\frac{11}{4}$$

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