Calculating the Rate of Decay

Decay measures how quickly something disappears or dies. Decay is often used to quantify the exponential decrease of bacteria or nuclear waste. In order to calculate exponential decay, you need to know the initial population and final population. Exponential decay occurs when the amount of decrease is directly proportional to how much exists.

1. Divide The Final Count by The Initial Count

Divide the final count by the initial count. For example, if you had 100 bacteria to start and 2 hours later had 80 bacteria, you would divide 80 by 100 to get 0.8.

2. Use Natural Log

Use the calculator to take the natural log (often abbreviated “ln” on calculators) of the result from the previous step. In this example, you would take the natural log of 0.8, which equals -0.223143551.

3. Divide the Result By Time

Divide the result from the last step by the number of time periods to find the rate of decay. In this example, you would divide -0.223143551 by 2, the number of hours, to get a rate of decay of -0.111571776. As the time unit in the example is hours, the decay rate is -0.111571776 per hour.

TL;DR (Too Long; Didn’t Read)

The minus sign in the result indicates a negative growth, or decay. To find the amount for any time period, multiply the time period by the decay rate and raise e, the natural logarithm base, to the power of the result. Then take that answer and multiply it by the initial value. For example, to find the bacteria population after 5 hours, multiply 5 by -0.111571776 to obtain -0.55785888. E to the power of -0.55785888 is 0.57243340. Multiply 0.57243340 by 100, the initial population, to get 57.243340 after 5 hours.