Modelling Exponential Decay - Using Logarithms
A common example of exponential decay is radioactive decay.
Radioactive materials, and some other substances, decompose according to a formula for exponential decay. A=A0ekt where k < 0.
That is, the amount of radioactive material A present at time t is given by the formula
A radioactive substance is often described in terms of its half-life, which is the time required for half the material to decompose.
After 500 years, a sample of radium-226 has decayed to 80.4\% of its original mass. Find the half-life of radium-226.
Let A= the mass of radium present at time t (t=0 corresponds to 500 years ago). We want to know for what time t is A = (1/2)A0. However, we do not even know what k is yet. Once we know what k is, we can set A in the formula for exponential decay to be equal to (1/2)A0, and then solve for t.
First we must determine k. We are given that after 500 years, the amount present is 80.4% of its original mass. That is, when t=500, A=0.804 A0. Substituting these values into the formula for exponential decay, we obtain:
Dividing through by A0 gives us 0.804 = e500k
which is an exponential equation.
To solve this equation, we take natural logs (ie. ln) of both sides. (Common logs could be used as well.)
ln ( 0.804) = ln (e500k)
We know that ln (e500k) = 500k by the cancellation properties of ln and e (see Logarithms). So the equation becomes ln ( 0.804) = 500k
and k= (ln 0.804)/500.
This is the exact solution; evaluate the natural log with a calculator to get the decimal approximation k = -0.000436 .
Since we now know k, we can write the formula (function) for the amount of radium present at time t as A=A0 e-0.000436 t.
(1/2)A0=A0 e-0.000436 t
Now, we can finally find the half-life. We set A=1/2 A0 and solve for t.
Dividing through by A0 again, we get:
1/2 = e-0.000436 t.
To solve for t, take natural logs: ln(1/2) = ln[e-0.000436 t].
Then applying the cancellation property for logarithms yields ln (1/2) = -0.000436 t
So t= ln(1/2) /(-0.000436)
or t = 1590.
The half-life is approximately 1590 years.
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