Solve 0=e^-x-1/2

$$0 = e ^ { - x } - \frac { 1 } { 2 }$$

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.

$$e^{-x}-\frac{1}{2}=0$$

Add \frac{1}{2} to both sides of the equation.

$$e^{-x}=\frac{1}{2}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-x})=\log(\frac{1}{2})$$

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

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

Divide both sides by \log(e).

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

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

$$-x=\log_{e}\left(\frac{1}{2}\right)$$

Divide both sides by -1.

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

Solve for x (complex solution)

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

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