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Application of Logarithms – finding the half life

Finding the half life

The half life is the time it takes to decay to half the amount. Thus, we want to know the time it takes to decay from \(10\) kg to \(5\) kg. Using the equation from the previous page:

\[ M= 10 e^{-2.043302495\times t}\]
we have:
\begin{eqnarray*}
M &=& 10 e^{-2.043302495\times t} \\
5&=& 10 e^{-2.043302495\times t} \\
\frac{1}{2} &=& e^{-2.043302495\times t} \\
\ln \left(\frac{1}{2}\right) &=& -2.043302495\times t \\
t&=& \frac{\ln(0.5)}{-2.043302495} \\
&\approx& 0.339 \mbox{ years} \\
\end{eqnarray*}

Therefore, the half life is approximately \(0.3\) years.