Exponential Functions

Significance

Exponential functions are functions of the form f(x) = b^x for a fixed base b which could be any positive real number. Exponential functions are characterized by the fact that their rate of growth is proportional to their value. For example, suppose we start with a population of cells such that its growth rate at any time is proportional to its size. The number of cells after t years will then be a^t (an exponential function) for some a>0.

Standard Notation

One exponential function, f(x)=e^x, is distinguished among all exponential functions by the fact that its rate of growth at x is exactly equal to the value e^x of the function at x. The number "e" is named after Leonid Euler.

Rules of Exponentiation

The main rules used in manipulating exponential functions are:

b^x+y = (b^x)(b^y)
b^xy = (b^x)^y
b⁰ = 1
b^-x = 1/(b^x)

See also the rules of exponentiation discussed in the page on power functions.

Graphs of Exponential Functions

The shape of the graph of y = b^x depends on whether b < 1, b = 1, or b > 1 as shown on the right. The red graph is the graph of b^x (b > 1), the blue graph is the graph of 1^x, and the green graph is the graph of (1/b)^x (b < 1).

Since

b^x = (a ^log_ab)^x = a^(log_ab)x,

different exponential functions are really the same with a scaling of the x-axis. If the scaling is negative, the x-axis is flipped, which accounts for the different behavior.

Exponential Growth and Decay

A function whose rate of change is proportional to its value exhibits exponential growth if the constant of proportionality is positive and exponentional decay if the constant of proportionality is negative. For exponential growth, the function is given by kb^x with b > 1, and functions governed by exponential decay are of the same form with b < 1. Populations might exhibit exponential growth in the absence of constraints, while quantities of a radioactive isotope exhibit exponential decay.

If b > 1, the exponential function f(x) = b^x grows faster than any polynomial (or rational) function. In other words, if g(x) is a polynomial, there is some positive number M such that f(x) > g(x) for every x > M. Similarly, if b < 1, the function b^x has zero has a horizontal asymptote for large positive x and it nears this asymptote faster than any rational function. Thus, (1.00001)^x is eventually much, much bigger than x¹⁰⁰⁰ and (.99999)^x is eventaully much smaller than x^-1000.

Exponential Equations

These are done on the nex page.