Solve 8=4^3x+3

$$8 = 4 ^ { 3 x } + 3$$

Solve for x

$$x=\frac{\log_{2}\left(5\right)}{6}\approx 0.386988016$$

Steps Using Definition of Logarithm

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

$$4^{3x}+3=8$$

Subtract 3 from both sides of the equation.

$$4^{3x}=5$$

Take the logarithm of both sides of the equation.

$$\log(4^{3x})=\log(5)$$

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

$$3x\log(4)=\log(5)$$

Divide both sides by \log(4).

$$3x=\frac{\log(5)}{\log(4)}$$

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

$$3x=\log_{4}\left(5\right)$$

Divide both sides by 3.

$$x=\frac{\log_{2}\left(5\right)}{2\times 3}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{3\ln(2)}+\frac{\log_{2}\left(5\right)}{6}$$

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