Solve 0.25={(e)^-2x}

$$0.25= {(e)^{ -2x }}$$

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.

$$e^{-2x}=0.25$$

Take the logarithm of both sides of the equation.

$$\log(e^{-2x})=\log(0.25)$$

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

$$-2x\log(e)=\log(0.25)$$

Divide both sides by \log(e).

$$-2x=\frac{\log(0.25)}{\log(e)}$$

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

$$-2x=\log_{e}\left(0.25\right)$$

Divide both sides by -2.

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

Solve for x (complex solution)

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

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