Exponential Functions | Research & Encyclopedia Articles

Exponential Functions

Exponential functions have the form f(x)= ab^xwhere a is a non-zero real number, b is a positive real number not equal to 1, and x can take on all real number values. Examples of exponential functions include f(x) = 2^x, g(x) = 3(0.5)^x, and h(t) = 1300(1.02)^t. In general, the graph of f(x) = ab^x has a y-intercept at (0,a) and is asymptotic to the x-axis from above if a is positive and from below if a is negative. If b is greater than 1, then the graph is asymptotic to the x-axis as x decreases without bound and increases without bound as x increases without bound. If b is less than 1, then the graph is asymptotic to the x-axis as x increases without bound and increases without bound as x decreases without bound.

Exponential functions have a wide range of applications in finance, economics, biology, ecology, physics, and other sciences. A simple example is a bank account in which a certain initial amount of money is deposited at a given interest rate and we wish to track the growth of the money in this account over some specified time period. For example, if $1000 is deposited in an account earning 5% interest compounded annually, then the exponential function f(t) = 1000(1.05) ^t gives the amount of money in the account after t years. So at the end of 10 years we would calculate 1000(1.05)¹⁰ = $1628.89. Scientists from various fields use exponential functions as models for growth and decay phenomena in which a quantity is assumed to grow or decay at a rate which is proportional to the amount of the quantity at any given time. This assumption leads to a differential equation of the form dq/dt = kq(t) where q(t) is the amount of the quantity present at time t. In calculus courses it is shown that this differential equation has a solution of the form q(t) = q(0)e^kt where e is the base of the so-called natural logarithm system and is approximately 2.71828. The equation q(t) = q(0)e^kt is, of course exponential. When k is positive, the equation models growth; when k is negative, it models decay. As an example, suppose that a biologist is studying the growth of a certain colony of bacteria. She estimates that 1000 bacteria are present at her initial observation, and that an hour later the population has tripled. The equation she can use to model this growth has the form q(t) = 1000e^kt where k can be determined by using the fact that the population tripled in one hour. In this case k = ln(3), the natural logarithm of 3, and the simplest form of the exponential model is q(t) = 1000(3)^t. So the biologist could predict that after 5 hours the population would be approximately 1000(3)⁵ = 243,000. Physicists use exponential functions to study the decay of radioactive substances. Such functions can be used to approximate the age of ancient archeological organisms by estimating the amount of, say, radioactive carbon-14 present today in the object, knowing the amount of C-14 which would have been in the object when it was alive, and using the exponential model to solve for the number of years since the organism was alive.

The power of exponential growth can be illustrated by the following simple example. Suppose you are taking on a certain job for one month and you are given the choice of being paid a fee of $20,000 for the month or being paid 1 penny on the first day, 2 pennies on the second day, 4 pennies on the third day, 8 pennies on the fourth day, and so on with the number of pennies earned doubling each day through the end of the month. A person who does not understand the nature of exponential growth might opt for the flat $20,000 fee. However, the exponential function y = 2^x-1 will give the number of pennies earned on day x. So on the 31st day of the month, you would be paid 2³⁰ pennies or $10,737,418.24! That's right, more than ten million dollars! Note that this is just the amount paid on the last day of the month and does not include the total from the previous 30 days! In a more practical example, suppose you invest $1000 in a stock market fund which has historically earned 12% per year, about average for the stock market. In 30 years, you would have $1000(1.12)³⁰ = $29,959.92 and that's if you added nothing else to your account during the 30 years.

Exponential functions and modifications of exponential functions are currently being used to study such phenomena as the growth of the internet, the spread of AIDS, the projected growth or decay of the national debt, and much more. Wherever growth and decay are studied, exponential models are sure to be a fundamental part of the study.