Solve 8=e^3x

$$8 = e ^ { 3 x }$$

Solve for x

$$x=\ln(2)\approx 0.693147181$$

Steps Using Definition of Logarithm

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

$$e^{3x}=8$$

Take the logarithm of both sides of the equation.

$$\log(e^{3x})=\log(8)$$

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

$$3x\log(e)=\log(8)$$

Divide both sides by \log(e).

$$3x=\frac{\log(8)}{\log(e)}$$

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

$$3x=\log_{e}\left(8\right)$$

Divide both sides by 3.

$$x=\frac{3\ln(2)}{3}$$

Solve for x (complex solution)

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

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