Solve 4=2*e^2x

$$4 = 2 \cdot e ^ { 2 x }$$

Solve for x

$$x=\frac{\ln(2)}{2}\approx 0.34657359$$

Steps Using Definition of Logarithm

Divide both sides by 2.

$$\frac{4}{2}=e^{2x}$$

Divide 4 by 2 to get 2.

$$2=e^{2x}$$

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

$$e^{2x}=2$$

Take the logarithm of both sides of the equation.

$$\log(e^{2x})=\log(2)$$

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

$$2x\log(e)=\log(2)$$

Divide both sides by \log(e).

$$2x=\frac{\log(2)}{\log(e)}$$

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

$$2x=\log_{e}\left(2\right)$$

Divide both sides by 2.

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

Solve for x (complex solution)

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

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