Solve e^2x=2/5

$$e ^ { 2 x } = \frac { 2 } { 5 }$$

Solve for x

$$x=\frac{\ln(\frac{2}{5})}{2}\approx -0.458145366$$

Steps Using Definition of Logarithm

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

$$e^{2x}=\frac{2}{5}$$

Take the logarithm of both sides of the equation.

$$\log(e^{2x})=\log(\frac{2}{5})$$

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

$$2x\log(e)=\log(\frac{2}{5})$$

Divide both sides by \log(e).

$$2x=\frac{\log(\frac{2}{5})}{\log(e)}$$

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

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

Divide both sides by 2.

$$x=\frac{\ln(\frac{2}{5})}{2}$$

Solve for x (complex solution)

$$x=\frac{\ln(\frac{2}{5})}{2}+i\pi n_{1}$$

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