Solve 27^3x-2=81

$$27 ^ { 3 x - 2 } = 81$$

Solve for x

$$x = \frac{10}{9} = 1\frac{1}{9} \approx 1.111111111$$

Steps Using Definition of Logarithm

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

$$27^{3x-2}=81$$

Take the logarithm of both sides of the equation.

$$\log(27^{3x-2})=\log(81)$$

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

$$\left(3x-2\right)\log(27)=\log(81)$$

Divide both sides by \log(27).

$$3x-2=\frac{\log(81)}{\log(27)}$$

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

$$3x-2=\log_{27}\left(81\right)$$

Add 2 to both sides of the equation.

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

Divide both sides by 3.

$$x=\frac{\frac{10}{3}}{3}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{9\ln(3)}+\frac{10}{9}$$

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