http://www.ck12.org Chapter 6. Transcendental Functions
6.4 Exponential Growth and Decay
Learning Objectives
A student will be able to:
- Apply the laws of exponential and logarithmic functions to a variety of applications.
- Model situations of growth and decay in a variety of problems.
When the rate of change in a substance or population is proportional to the amount present at any timet, we say that
this substance or population is going through either a decay or a growth, depending on the sign of the constant of
proportionality.
This kind of growth is calledexponential growthand is characterized by rapid growth or decay. For example,
a population of bacteria may increase exponentially with time because the rate of change of its population is
proportional to its population at a given instant of time (more bacteria make more bacteria and fewer bacteria make
fewer bacteria). The decomposition of a radioactive substance is another example in which the rate of decay is
proportional to the amount of the substance at a given time instant. In the business world, the interest added to an
investment each day, month, or year is proportional to the amount present, so this is also an example of exponential
growth.
Mathematically, the relationship between amountyand timetis a differential equation:
dy
dt=ky.Separating variables,
dy
y =kdt,and integrating both sides,
∫ dy
y =∫
kdt,gives us
lny=kt+C,
y=ekteC
=Cekt.