# How can half-life be calculated?

Nov 1, 2015

If the decay is exponential, i.e. $x = {x}_{0} {e}^{- k t}$, then the half-life is given by $\ln \frac{2}{k}$.

#### Explanation:

Let $\tau$ be the half-life. $\tau$ is the time taken for ${x}_{0}$ to reach ${x}_{0} / 2$

${x}_{0} / 2 = {x}_{0} {e}^{- k \tau}$

$\frac{1}{2} = {e}^{- k \tau}$

$2 = {e}^{k \tau}$

$\ln \left(2\right) = k \tau$

$\tau = \ln \frac{2}{k}$