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If $375$ mg of a radioactive substance decays to $300$ mg in $72$ hours, find the

half-life of the element. I first used the mathematical formula of $$A = A_0e^{kt}$$

or exponential decay. After doing my calculations with natural logarithms, I got

an odd answer of $1.3$ days, by dividing $\ln (375)/\ln (72)$. I know this cannot be the half-life of a radioactive element. Where is my wrong step?

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1 Answer 1

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Because it is exponential decay, I would prefer to use $A(t)=A(0)e^{-kt}$.

Then $\frac{300}{375}=e^{-72k}$, so $k=\frac{1}{72}\ln(375/300)$.

Now for the half-life $\tau$, we have $\frac{1}{2}A(0)=A(0)e^{-k\tau}$, so $k\tau=\ln 2$. It follows that the half-life, in hours, is $72\frac{\ln 2}{\ln(375/300)}$.

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