Solve 81=3^x-2

$$81 = 3 ^ { x - 2 }$$

Solve for x

$$x=6$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$3^{x-2}=81$$

Take the logarithm of both sides of the equation.

$$\log(3^{x-2})=\log(81)$$

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

$$\left(x-2\right)\log(3)=\log(81)$$

Divide both sides by \log(3).

$$x-2=\frac{\log(81)}{\log(3)}$$

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

$$x-2=\log_{3}\left(81\right)$$

Add 2 to both sides of the equation.

$$x=4-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(3)}+6$$

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