Solve e^-x=0.5

$${ e }^{ -x } =0.5$$

Solve for x

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

Steps Using Definition of Logarithm

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

$$e^{-x}=0.5$$

Take the logarithm of both sides of the equation.

$$\log(e^{-x})=\log(0.5)$$

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

$$-x\log(e)=\log(0.5)$$

Divide both sides by \log(e).

$$-x=\frac{\log(0.5)}{\log(e)}$$

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

$$-x=\log_{e}\left(0.5\right)$$

Divide both sides by -1.

$$x=-\frac{\ln(2)}{-1}$$

Solve for x (complex solution)

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

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