Atoms and Nuclei Part 2

As we know that,
N=N0e−λt
where,
N= number of radioactive atoms left undecayed,
N0= original number of radioactive atoms,
t= elapsed time
and λ= decay constant.
⇒‌‌N1=N0e−λ⋅t1....(i)
and N2=N0e−λt2....(ii)
Consider Eq. (i), N1=N0e−λt1
⇒ ‌

N1

N0

=e−λt1⇒0.67=e−λt1
⇒‌‌log(0.67)=log(e−λt1)
⇒‌‌log‌0.67=−λt1.....(iii)
Consider Eq. (ii), N2=N0e−λt2
⇒‌‌‌

N2

N0

=e−λt2⇒0.33=e−λt2
⇒‌‌log(0.33)=log(e−λt2)
⇒‌‌log(0.33)=−λt2.....(iv)
Subtracting Eq. (iv) from Eq. (iii), we get
log‌0.67−log‌0.33=−λt1+λt2
⇒‌‌t2−t1=‌

log‌2

λ

=t1/2
∴ Half-life =t1/2=20min