The activity of a sample of a radio active material is A₁ at time  $t_1 \mathrm{and} {\mathrm{A}}_2$  at time ${\mathrm{t}}_2({\mathrm{t}}_2>{\mathrm{t}}_1)$. If its mean life T, then

$$\begin{gathered} \mathrm{A}={\mathrm{A}}_0{\mathrm{e}}^{-\mathrm{\lambda t}}={\mathrm{A}}_0{\mathrm{e}}^{-\mathrm{t}/\mathrm{\tau }} ; \mathrm{where} \mathrm{\tau }=\mathrm{mean} \mathrm{life} \\ \mathrm{So} {\mathrm{A}}_1={\mathrm{A}}_0{\mathrm{e}}^{-{\mathrm{t}}_1/\mathrm{\tau }}\Rightarrow {\mathrm{A}}_0=\frac{{\mathrm{A}}_1}{{\mathrm{e}}^{-{\mathrm{t}}_1/\mathrm{T}}}={\mathrm{A}}_1{\mathrm{e}}^{t_1/\mathrm{\tau }} \\ \therefore {\mathrm{A}}_2={\mathrm{A}}_0{\mathrm{e}}^{-t_2/\mathrm{T}}=\left({\mathrm{A}}_1{\mathrm{e}}^{{\mathrm{t}}_1/\mathrm{T}}\right){\mathrm{e}}^{-{\mathrm{t}}_2/\mathrm{T}}\Rightarrow {\mathrm{A}}_2={\mathrm{A}}_1{\mathrm{e}}^{({\mathrm{t}}_1-{\mathrm{t}}_2)/\mathrm{T}} \Rightarrow \mathrm{T} = \frac{{\mathrm{t}}_1-{\mathrm{t}}_2}{\ln \left({\displaystyle \frac{{\mathrm{A}}_2}{{\mathrm{A}}_1}}\right)} \end{gathered}$$