10. The half life of a radioactive substance is 2.3 years. Scientists figure that it will be safe to handle it if there is only 10% of the radioactivity left. How long would one have to wait until it was deemed safe?

The formula for the amount of material left in radioactive decay is

Where *A* is the amount that is left, *A*_{o} is the original amount, *t* is the number of years, and *h* is the half life. Substitute the numbers into the equation, where *A* is 0.1*A*_{o}.

Cancel the *A*_{o}'s

and we have an exponential equation where the only unknown is the time. Take logs of both sides. It is customary to use natural logs for these types of problems. The main concern is that we will have a button for logarithms in that base on our calculator, and if a calculator has only one logarithm button, it will be a natural logarithm button.

Logarithms turn exponents into factors

and we get a linear equation. The only step is to move the constant factors with the unknown to the other side of the equation.

which gives us something we can punch up on our calulators.

2.3 ln (0.1)/ln (0.5) = 7.64043461

Does this make sense? (1/2)^{3} = 1/8, (1/2)^{4} = 1/16, so we will need between 3 and 4 half lives (closer to 3 than to 4) before we wind up with 1/10 of the original amount. 3x2.3 = 6.9, so this sounds about right.