An exponential function is a function of the following form: f(x)=bx where b (the base) is equal to some value greater than 0. The domain of an exponential function is the set of all real numbers and the range is the set of all positive real numbers ("Exponential Function Reference", 2018). Of particular note is the exponential function of base e, f(x)=ex, which is referred to as the natural exponential function or simply the exponential function (Maor, 1994).
Figure: The graph above shows the general shape of an exponential function.
Exponential functions are very interesting. They lack many of the common features of functions including x-intercepts, vertical asymptotes, maximums, minimums, and inflection points (Maor, 1994). However, an exponential function has one defining characteristic that sets it apart from all other functions: its rate of change. Because this property plays such a critical role in the application of exponential functions, we will discuss it in more depth in a section below.
Interesting Fact:
One of the popular problems in mathematics following the invention of calculus was the problem of the catenary, or hanging chain. Jacob Bernoulli first proposed this problem in the May 1690 issue of Acta eruditorium. Bernoulli wrote, "And now let this problem be proposed: To find the curve assumed by a loose string hung freely from two fixed points" (Maor, 1994).
Figure: Gateway Arch in St. Louis, Missouri
In June 1691, the Acta eruditorium published three correct solutions found by Huygens, Leibniz, and Johann Bernoulli (Jacob's brother). The curve's equation is not a parabola as one might first suppose. Rather, the catenary is given by the following equation in modern notation: y= (eax +e-ax)/2ax where a is a constant that depends on the parameters of the chain. Gateway Arch in St. Louis, Missouri is the shape of an inverted catenary (Maor, 1994).
Inverse of the Exponential Function
Whereas a function y=f(x) maps every x-value in the domain to a unique y-value, the inverse function takes every y-value and maps it back to the original x-value (which sometimes requires a domain restriction in order to be a function). The inverse function of f(x) is denoted by f-1(x). Using mathematical notation, the above description of the relationship between f(x) and its inverse can be written as:
f(f-1(x))=x and f-1(f(x))=x
Additionally, due to this relationship, the graphs of f(x) and f-1(x) are symmetric across the line y=x ("Inverse Functions", 2017).
Figure: The graphs of y=ex and y=ln(x) are symmetric across the line y=x.
As shown in the graph above, the exponential function is the inverse of the logarithmic function. Prior to Euler's time, the exponential function was merely thought of as the reverse of the logarithmic function (Maor, 1994). However, in his book Introductio in analysin infinitorum (1748), Euler placed the two functions on equal ground by giving them the following independent definitions (Maor, 1994):
ex = limx → ∞ (1+x/n)n and ln(x) = limx → ∞ n(x1/n -1)
Explore Euler's limit definition of the function y=ex in the applet below created by Arnaud Crouzet (2011). Note how closely the approximation (green) fits the graph of y=ex (black) as you drag the slider for the number of approximation steps (n). What do you think will happen as the number of approximation steps (n) approaches infinity?
In Introductio in analysin infinitorum (1748), Euler also emphasized our modern definition of logarithms, namely:
logb(y)=x if and only if y=bx
This definition further illustrates how exponential functions and logarithmic functions are inverses.
During the 1600s, a new branch of mathematics, known as calculus, was born. The invention of calculus is attributed to both Newton and Leibniz, who both worked independently on its development during a similar time period (Maor, 1994).
Figure: Isaac Newton, 1642-1727
Figure: Gottfried Wilhelm Leibniz, 1646-1716
There are two main divisions in calculus: differential calculus and integral calculus. We will focus on differential calculus in this section, or the study of the rate of change of a variable (Maor, 1994). If you are unfamiliar with the concept of derivatives, watch the video below created by Sal Khan for a brief explanation.
As we mentioned above, the defining characteristic of an exponential function is its rate of change, or its derivative. Using the applet below created by Rob Morris (2011), explore the first and second derivatives of different exponential functions. Use the slider at the top to change the value of the base or click on the buttons below the slider to see the graphs of special bases. In particular, check out the derivative of the exponential function of base e.
From your explorations with the applet above, note that the derivative of ex is itself. That is, the rate of change of the function f(x)=ex at a given point is the same as its value at that same point. Even more interesting, ex is the only function whose derivative is itself (with the exception of f(x)=0, but that result is not surprising or very interesting). Robert Courant and Herbert Robbins stated the following about this property: "The natural exponential function is identical with its derivative. This is the source of all the properties of the exponential function and the basic reason for its importance in applications" (Maor, 1994).
To view a proof of the derivative of ex is ex, check out the video below created by Sal Khan.
