Solve 6=2^3x+1

$$6= { 2 }^{ 3x+1 }$$

Solve for x

$$x=\frac{\log_{2}\left(3\right)}{3}\approx 0.528320834$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$2^{3x+1}=6$$

Take the logarithm of both sides of the equation.

$$\log(2^{3x+1})=\log(6)$$

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

$$\left(3x+1\right)\log(2)=\log(6)$$

Divide both sides by \log(2).

$$3x+1=\frac{\log(6)}{\log(2)}$$

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

$$3x+1=\log_{2}\left(6\right)$$

Subtract 1 from both sides of the equation.

$$3x=\log_{2}\left(6\right)-1$$

Divide both sides by 3.

$$x=\frac{\log_{2}\left(6\right)-1}{3}$$

Solve for x (complex solution)

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

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