Exponentials and logarithms

Summary: Exponential and logarithmic functions are used to study and analyze a variety of mathematical relationships.

Much of the language and notation of mathematics involves a very advanced shorthand. As ideas grow and become more complex, mathematicians seek ways to express highly condensed thought in relatively simple terms. Exponents are an elementary example: if one wants to multiply the number 2 times itself 10 times, rather than write “2•2•2•2•2•2•2•2•2•2” one can write “2¹⁰” instead. From these beginnings, which date to ancient Egypt and Babylon, the remarkable worlds of exponential and logarithmic functions emerge. When one develops the understanding of what it means to take 2 to any real number power, one naturally considers the function f(x)=2x, an example of what is called an “exponential function.” For larger and larger positive x, the function grows amazingly fast: 2¹⁰=1024, 2²⁰=1,048,576, and 2³⁰=1,073,741,824.

The exponential functionf(x)=e^x, where e is the so-called “natural base,” an irrational number whose decimal approximation is e = 2.71828, is an important exponential function. With e in homage to the great Swiss mathematician Leonhard Euler (1707-1783), this special exponential function f(x)=e^x might rightly lay claim to the title of “the most important function in all of mathematics.” Exponential growth and decay functions, along with the number e itself, have a wide variety of uses and applications.

In classrooms in the twenty-first century, the logarithm of a number is defined as the exponent or power to which a stated number, called the “base,” is raised to obtain the given number. The development of logarithms in the seventeenth century led to a revolution in scientific calculation, especially when the slide rule replaced tables of logarithms. While the advent of calculators and computers eliminated the need for calculation by logarithms in the latter part of the twentieth century, logarithms remain important in order to understand financial and natural processes. For instance, the Richter scale to measure earthquakes, named for Charles Richter, is a logarithmic scale. In chemistry, the pH scale is based on the negative logarithm of the concentration of free hydrogen ions. Students in the middle grades investigate exponential notation while high school students explore exponential and logarithmic functions.

Archimedes of Syracuse investigated that the addition of what he called “orders” corresponded with their product, known today as the “first law of exponents.” The number e may have first appeared in the early seventeenth century in an appendix to John Napier’s work on logarithms. This number also arose in the work of Christiaan Huygens in the mid-seventeenth century when he was exploring the area under the hyperbola xy=1. Finally, in the late seventeenth century through work involving continuous compound interest, Jacob Bernoulli was led to consider the expression

for large values of n, and this expression approaches e as n grows without bound. Mathematicians explored many issues related to e and exponentials, including such people as Euler, Gotthold Eisenstein, and others, who investigated the convergence of sequences of iterated exponentials. Bernoulli may also have been the first mathematician to realize that the number e was intricately linked to emerging ideas with logarithms.

The Natural Exponential Function

Because any exponential function can be written in terms of e, one finds that functions of the form P(x)=Me^kx, where M and k are constants that depend on the context, arise in many natural settings. Exponential cell and population growth, as well as exponential decay in radioactive materials, are modeled by functions of this form. Once the values of M and k are identified, the function easily indicates the corresponding output for any input value x. For example, if a car is initially valued (at time t=100) at $10,000 that depreciates at a certain continuous rate, one might use the function P(t)=10000e^-0.2t to model the worth of the car in year t.

Functions like this generate very natural questions, including ones like “At what time t will the car’s value be $3,000?” Before trying to answer this more complicated question, consider some simpler ones. For instance, what value of t makes 10^t=17? Since 10¹=10, while 10²=100, it seems like there ought to be a number between 1 and 2 such that 10 raised to that power is 17. But what is the number? Here, some very considerable mathematical ideas are involved: the function y(t)=10^t is continuous; the range of y is all positive real numbers; and −y is always increasing, making it a one-to-one function. All these facts together combine to indicate that one can pick any positive real number y and know that there must be one and only one real number t that satisfies the equation 10^t=y. In other words, there is a function h that takes any positive real number y, and to this value y associates the real number t so that 10 raised to the power t is y. This explanation is how teachers usually describe to students where logarithms come from the logarithm is the very function that accomplishes this association. It is all a matter of perspective; if t is known and y is sought, the exponential function is used, while if y is known and t is sought, the logarithm function is used. Expressed in words, it is “y equals 10 to the power t” and “t is the power to which we raise 10 to get y.” Babylonian clay tablets presented similar questions.

Historical Development

Historically, the further development of logarithms arose very differently. In the late fifteenth century and early sixteenth century, both John Napier and Jost Burgi, who were each interested in key problems in astronomy, developed logarithms for a much different use: as a new tool to help do arithmetic with large numbers. Their approach to logarithms was fundamentally geometric, as algebra was not yet sufficiently well developed to aid their work, although Napier’s approach was more algebraic than Burgi’s methods. Napier noted, “Seeing there is nothing that is so troublesome to mathematical practice, nor doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.” In 1624, Henry Briggs published logarithm tables in Arithmetica Logarithmica and he is noted by some as perhaps the man most responsible for popularizing logarithms among scientists. The development of the slide rule made logarithms easy to use, since they reduced the reliance on tables. In 1620, Edmund Gunter noted logarithms on a ruler by marking the position of numbers relative to their logarithms. William Oughtred placed two sliding logarithmic rulers next to each other and by 1630, the portable circular slide rule reduced multiplication computations to the act of lining up two numbers and reading a scale. Logarithms remain a useful way to deal with large numbers in the early twenty-first century, because the logarithm of a large number is a much, much smaller one. R. C. Pierce Jr. noted, “It has been postulated that logarithms literally lengthened the life spans of astronomers who had formerly been sorely bent and often broken early by the masses of calculations their art required.” Modern mathematicians have also come to fully understand the connection between logarithms and the area under the curve xy = 1, which was explored by Huygens in the 1600s.

Using Logarithms to Solve Exponential Functions

Perhaps the most powerful property of logarithms is that they “undo” exponential functions. For example, for the natural logarithm of base e, denoted “ln,” one obtains ln(e⁵)=5. Remember, ln(e⁵) means “the power to which one raises e to get e⁵.” This power, of course, is 5. The general property that holds here is that for any real number t, ln(e^t)=t. This rule proves to be immensely useful in solving exponential equations. To see how, consider an earlier example: the function P(t)=10000e^-0.2t (the value of a car in year t). At what time t will the car’s value be $3,000? This question is equivalent to solving the equation:

Taking the natural logarithm of both sides of the equation “undoes” the effects of the exponential function and hence gains more direct access to the variable t: ln(e^-0.2t)=-0.2t.

Dividing both sides of the last equation above by −0.2, one finds that

so that the car’s value will be $3,000 in just over six years. The natural logarithm of 0.3 is central to answering the question.

While the motivation for the need for logarithms can be seen in relatively elementary terms solving exponential equations the actual mathematics that explains what logarithms really are and how they work is deep and is best supported using some sophisticated ideas from calculus. Even with exponential functions, there are some big questions without answers: how is e to the 5th power calculated? How is the natural logarithm of 0.3 computed? Until the invention of personal computers in the 1970s, such computations were all done by hand, usually with the assistance of elaborate tables, or with slide rules. At one point in history, entire books were written that held nothing but tables of values for logarithms. People now use inexpensive hand-held calculators, computer algebra systems like Maple or Mathematica, or even Google, and each returns a value almost immediately. These modern technological tools rely on a rich and beautiful mathematical theory of exponential and logarithmic functions. Beyond their interesting mathematical properties, exponential and logarithmic functions remain important for their many applications, such as the key role that exponential functions play in the study of differential equations, including those that model vibrations in bridges and buildings, thus forming a central component of modern civil engineering.

Bibliography

Maor, Eli. e: The Story of a Number. Reprint. Princeton, NJ: Princeton University Press, 1994.

Nahin, Paul J. An Imaginary Tale: The Story of i (The Square Root of Minus One). Princeton, NJ: Princeton University Press, 2010

Pierce Jr., R. C. “A Brief History of Logarithms.” The Two-Year College Mathematics Journal 8, no. 1 (1977).

Stoll, Cliff. “When Slide Rules Ruled.” Scientific American 294, no. 5 (2006).

Strogatz Steven. “Power Tools NYTimes.com” http://opinionator.blogs.nytimes.com/2010/03/28/power-tools.