- Use like bases to solve exponential equations.
- Use logarithms to solve exponential equations.
- Solve logarithmic equations
- Solve applied problems involving exponential and logarithmic equations.
Solve applied problems involving exponential and logarithmic equations
We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.
One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life. The table below lists the half-life for several of the more common radioactive substances.
| Substance | Use | Half-life |
|---|---|---|
| gallium-67 | nuclear medicine | 80 hours |
| cobalt-60 | manufacturing | 5.3 years |
| technetium-99m | nuclear medicine | 6 hours |
| americium-241 | construction | 432 years |
| carbon-14 | archeological dating | 5,715 years |
| uranium-235 | atomic power | 703,800,000 years |
We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:
radioactive decay
[latex]\begin{align}&A\left(t\right)={A}_{0}{e}^{\frac{\mathrm{ln}\left(0.5\right)}{T}t} \\ &A\left(t\right)={A}_{0}{e}^{\mathrm{ln}\left(0.5\right)\frac{t}{T}} \\ &A\left(t\right)={A}_{0}{\left({e}^{\mathrm{ln}\left(0.5\right)}\right)}^{\frac{t}{T}} \\ &A\left(t\right)={A}_{0}{\left(\frac{1}{2}\right)}^{\frac{t}{T}} \end{align}[/latex]
where
- [latex]{A}_{0}[/latex] is the amount initially present
- T is the half-life of the substance
- t is the time period over which the substance is studied
- y is the amount of the substance present after time t
How long will it take for ten percent of a 1000-gram sample of uranium-235 to decay?