Solve 9^3x-3=27

$$9 ^ { 3 x - 3 } = 27$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Definition of Logarithm

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

$$9^{3x-3}=27$$

Take the logarithm of both sides of the equation.

$$\log(9^{3x-3})=\log(27)$$

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

$$\left(3x-3\right)\log(9)=\log(27)$$

Divide both sides by \log(9).

$$3x-3=\frac{\log(27)}{\log(9)}$$

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

$$3x-3=\log_{9}\left(27\right)$$

Add 3 to both sides of the equation.

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

Divide both sides by 3.

$$x=\frac{\frac{9}{2}}{3}$$

Solve for x (complex solution)

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

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