THE HISTORY OF LOGARITHMS

In such conditions, it is hardly surprising that many mathematicians were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. In particular, the Scottish mathematician John Napier was famous for his devices to assist with computation. He invented a well-known mathematical artifact, the ingenious numbering rods more quaintly known as “Napier's bones,” that offered mechanical means for facilitating computation. (For additional information on “Napier's bones,” see the article, “John Napier: His Life, His Logs, and His Bones” (2006).) In addition, Napier recognized the potential of the recent developments in mathematics, particularly those of prosthaphaeresis, decimal fractions, and symbolic index arithmetic, to tackle the issue of reducing computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry. Therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant –

THE HISTORY OF EXPONENTS

History usually starts way back at the beginning and then relates developmental events to the present so you can understand how you got to where you are. With mathematics, in this case exponents, it will make much more sense to start with a current understanding and meaning of exponents and work backward to from where they came. First and foremost, let's make sure you understand what an exponent is because it can get quite complicated. In this case, we'll keep it simple.

1. Where We Are Nowo This is the junior high school version, so we should all understand this. An

exponent reflects a number multiplied by itself, like 2 times 2 equals 4. In exponential form that could be written 2², called two squared. The raised 2 is the exponent and the lower case 2 is the base number. If you wanted to write 2x2x2 it could be written as 2³ or two to the third power. The same goes for any base number, 8² is 8x8 or 64. You get it. You could use any number as the base and the number of times you want to multiply it by itself would become the exponent.

Where Did Exponents Come From?

o The word itself comes from Latin, expo, meaning out of, and ponere, meaning place. While the word exponent came to mean different things, the first recorded modern use of exponent in mathematics was in a book called "Arithemetica Integra," written in 1544 by English author and mathematician Michael Stifel. But he was working simply with a base of two, so the exponent 3 would mean the number of 2s you would need to multiply to get 8. It would look like this 2³=8. The way Stifel would say it is kind of backwards when compared to the way we think about it today. He would say "3 is the 'setting out' of 8." Today, we would refer the equation simply as 2 cubed. Remember, he was working exclusively

with a base or factor of 2 and translating from Latin a little more literally than we do today.

APPLICATIONS OF LOGARITHMS

Careers That Use LogarithmsLogarithms, the inverses of exponential functions, are used in many occupations. Perhaps the most well-known use of logarithms is in the Richter scale, which determines the intensity and magnitude of earthquakes. Yet, there are many other professionals who use logarithms in their careers. Anyone who calculates the quantity of things that increase or decrease exponentially uses logarithms. This includes engineers, coroners, financiers, computer programmers, mathematicians, medical researchers, farmers, physicists and archaeologists. Because there is no definitive list of careers that require the use of logarithms, below is a brief sampling of how some careers employ these logs.

1. Coronero You often see logarithms in action on television crime shows, according to

Michael Breen of the American Mathematical Society. On such shows, coroners often attempt to determine how long a body has been dead. These television coroners, as well as their real-life counterparts, use logarithms to make such determinations. Once a body dies, it begins to cool. To calculate how long the body has been dead, the coroner must know how long the body temperature has not been at 98.6 degrees. Because the rate of the body cooling is proportionate to temperature differences between the body and its surroundings, the answer is found by calculating exponential decay using logarithms.

Actuarial Science

o An actuary's job is to calculate costs and risks. Many of these calculations involve complicated statistics. For example, an actuary may work as a consultant designing pension plans for a company's employees.To do so, the actuary may have to figure out the chances of a particular 50-year-old employee living to be 89 years old. The actuary then designs that person's pension using statistics that are exponential in nature, and that's where the logarithms enter in.

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Medicine

o Logarithms are used in both nuclear and internal medicine. For example, they are used for investigating pH concentrations, determining amounts of radioactive decay, as well as amounts of bacterial growth. Logarithms also are used in obstetrics. When a woman becomes pregnant, she produces a hormone known as

human chorionic gonadotropin. Since the levels of this hormone increase exponentially, and at different rates with each woman, logarithms can be used to determine when pregnancy occurred and to predict fetus growth.

Archaeology

o Archaeologists use logarithms to determine the age of artifacts, such as bones and other fibers, up to 50,000 years old. When a plant or animal dies, the isotope of carbon, Carbon-14, decays into the atmosphere. Using logs, archaeologists can compare the decaying Carbon-14 to the Carbon-12, which remains constant in an organism even after death, to determine the age of the artifact. For example, this type of carbon dating was used to determine the age of the Dead Sea Scrolls.

 

Real World Application of Exponents

1. Carpentryo In carpentry, surface area is expressed in squared units. So the units have their

own exponents. For example, a board that is 10 feet long and 3 inches wide has a surface area on one side of 3/12 feet x 10 feet, or 2.5 ft^2. The 2 is the exponent of the unit “feet,” to indicate that the measurement is in two dimensions (area) instead of one dimension (length).

Compounding Interest

o Bank balances and loans earn interest. Compounded interest is interest that--after an account earns that interest--itself begins earning interest. So if an annual interest rate is 4%, and it earns the interest on a $100 balance for 3 years, an exponent represents the earning of new interest on the old principal for several years: $100x1.04x1.04x1.04 = $100x1.04^3.We Stock Over 500,000 Products. Free Delivery, Order Online Now!

Continuous Compounding

o It can be shown that e^x has the shape of the curve of continuously compounded interest, where e is an irrational number whose first few digits are 2.718281828. So $100 that is continuously compounded at an annual rate of 4% will equal $100xe^0.04 after one year. This equals $104.08. After 2 years, it will equal $100xe^(0.04x2).

In fact, many biological processes also follow the e^x curve shape, which makes sense, since biological processes tend to grow continuously instead of in periodic spurts (like the periodic compounding of a bank balance).

Scientific Notation

o Suppose you have a chemical concentration 0.000383 grams per liter, or a solar mass of 456,000,000,000,000,000,000 kg. Those zeroes take up a lot of space. Scientific notation simplifies this by writing the last number, for example, as 456x10^18 kg, which means 456 followed by 18 zeroes. Inversely, the chemical concentration is written with a negative sign: 3.83x10^-4. The -4 means that the decimal point is moved to the left four places.