Solve 6^3x+2=115

$$6 ^ { 3 x + 2 } = 115$$

Solve for x

$$x=\frac{\log_{6}\left(115\right)-2}{3}\approx 0.216065681$$

Steps Using Definition of Logarithm

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

$$6^{3x+2}=115$$

Take the logarithm of both sides of the equation.

$$\log(6^{3x+2})=\log(115)$$

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

$$\left(3x+2\right)\log(6)=\log(115)$$

Divide both sides by \log(6).

$$3x+2=\frac{\log(115)}{\log(6)}$$

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

$$3x+2=\log_{6}\left(115\right)$$

Subtract 2 from both sides of the equation.

$$3x=\log_{6}\left(115\right)-2$$

Divide both sides by 3.

$$x=\frac{\log_{6}\left(115\right)-2}{3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(6)}+\frac{\log_{6}\left(115\right)}{3}-\frac{2}{3}$$

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