EXAMPLE 5 Radioactivity in Your Home

The major radioactivity exposure for most Americans is from radon, which is formed from the decay of uranium. Radon-222 is a gas that enters many U.S. homes from the underlying soil and rock. It causes 10% to 15% of lung cancers in the United States (the rest are due to smoking). The half-life of radon-222 is 3.82 days. How long does it take for a quantity of radon-222 to decay to one-thousandth (0.1%) of the original amount?

For radon-222, we have days and want to determine the when :

Dividing both sides by gives

and taking the natural logarithm of each side gives

where the values and you get from your calculator. Solving for , we find