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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.