Solve (3^2)^2x-3=3^3

$$( 3 ^ { 2 } ) ^ { 2 x - 3 } = 3 ^ { 3 }$$

Solve for x

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

Steps Using Definition of Logarithm

Calculate 3 to the power of 2 and get 9.

$$9^{2x-3}=3^{3}$$

Calculate 3 to the power of 3 and get 27.

$$9^{2x-3}=27$$

Take the logarithm of both sides of the equation.

$$\log(9^{2x-3})=\log(27)$$

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

$$\left(2x-3\right)\log(9)=\log(27)$$

Divide both sides by \log(9).

$$2x-3=\frac{\log(27)}{\log(9)}$$

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

$$2x-3=\log_{9}\left(27\right)$$

Add 3 to both sides of the equation.

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

Divide both sides by 2.

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

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2\ln(3)}+\frac{9}{4}$$

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