Summary

Exponential functions are a family of functions with several unusual properties. In particular,

Exponential functions describe the growth of many natural systems, such as bacterial or human populations, the growth of your bank account, etc. They are generally written in the form

where a > 0 and . a is called the base of the exponential function. If

• a>1, then f is an increasing function with a horizontal asymptote of y=0 for values of x tending toward negative infinity; and
• 0<a<1, then f is a decreasing function with a horizontal asymptote of y=0 for values of x tending toward infinity.

The most important special value of the base a is a rather strange value called e, where .

The domain of an exponential function is all real numbers, but the range is only the positive numbers.

The example that we find the most useful for motivating the notion of exponential functions is the problem of compound interest. If we look at how interest is computed on a deposit of fixed principal P, we find that it depends on how often the interest is computed. Assume that the interest rate is r, and that t represents years. Then

Now it turns out that as m, the number of payments per year, increases without bound, the quantity

so that continually compounded interest bearing accounts are computed as

One property of exponential functions that is very important is that

That is, if the values of an exponential function are equal, then so are the arguments. This says that exponential functions are one-to-one.