Here, `tau_1=4xx10^(19)yrs,`
`tau_2=2xx10^(9)yrs,`
`(M_1)/(M_2)=100`
`(A_1)/(A_2)=1.02`
Let `N_1, N_2` be the number of atoms in the two isotope at time t
As M=NA
i.e. `N=M/A`
`:. (N_1)/(N_2)= (M_1)/(M_2). (A_1)/(A_2)=100/1.02....(i)`
for first isotope,
`N_1=(N_0)_1e^(-lambda_1 t)=(N_0)_1e^(-t//tau_(1))`
`N_2=(N_0)_2e^(-lambda_2 t)=(N_0)_2e^(-t//tau_(2 ))`
`(N_1)/(N_2)=e^(-t(1/(tau_1)-1/(tau_2)))=e^(t(1/(tau_1)-1/(tau_2)))=100/1.02.....(ii)`
form (i) and (ii),
`t(1/(tau_1)-1/(tau_2))=log_e (100/1.02)`
`:. t=(2.3026log_(10) (100//1.02)xxtau_2xxtau_1)/(tau_1-tau_(2))`
`=(2.3026 (2-0.0086)xx4xx10^9xx2xx10^(9))/((4xx10^9-2xx10^(9)))`
`=1.834xx10^(10)yrs`