The half-life ($T_{1/2}$) is the time taken for the number of radioactive nuclei to reduce to half of its initial value ($N=N_0/2$). Substituting this into the decay equation $N=N_0{e}^{-\lambda t}$ gives: $N_0/2=N_0{e}^{-\lambda T_{1/2}}$ $1/2=e^{-\lambda T_{1/2}}$ $\ln (1/2)=-\lambda T_{1/2}$ $T_{1/2}=\frac{\ln 2}{\lambda }\approx \frac{0.693}{\lambda }$.