Solve 2^(6x+3)=6

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

Solve for x

$$x=\frac{\log_{2}\left(3\right)-2}{6}\approx -0.069172917$$

Steps Using Definition of Logarithm

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

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

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(2).

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

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

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

Subtract 3 from both sides of the equation.

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

Divide both sides by 6.

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

Solve for x (complex solution)

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

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