Solve 3^3x+3=6561

$$3 ^ { 3 x + 3 } = 6561$$

Solve for x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps Using Definition of Logarithm

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

$$3^{3x+3}=6561$$

Take the logarithm of both sides of the equation.

$$\log(3^{3x+3})=\log(6561)$$

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

$$\left(3x+3\right)\log(3)=\log(6561)$$

Divide both sides by \log(3).

$$3x+3=\frac{\log(6561)}{\log(3)}$$

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

$$3x+3=\log_{3}\left(6561\right)$$

Subtract 3 from both sides of the equation.

$$3x=8-3$$

Divide both sides by 3.

$$x=\frac{5}{3}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{3\ln(3)}+\frac{5}{3}$$

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