Solve {3}^frac{1{3}x}=3^3

$${ 3 }^{ \frac{ 1 }{ 3 } x } = { 3 }^{ 3 }$$

Solve for x

$$x=9$$

Steps Using Definition of Logarithm

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

$$3^{\frac{1}{3}x}=27$$

Take the logarithm of both sides of the equation.

$$\log(3^{\frac{1}{3}x})=\log(27)$$

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

$$\frac{1}{3}x\log(3)=\log(27)$$

Divide both sides by \log(3).

$$\frac{1}{3}x=\frac{\log(27)}{\log(3)}$$

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

$$\frac{1}{3}x=\log_{3}\left(27\right)$$

Multiply both sides by 3.

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

Solve for x (complex solution)

$$x=\frac{i\times 6\pi n_{1}}{\ln(3)}+9$$

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