Logarithms and Fast Calculations Jeremy Renfro

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1 Logarithms and Fast Calculations Jeremy Renfro In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008

2 Logarithms and Fast Calculations This paper discusses logarithm functions and their historical role in fast calculations. Today, our point of view might be that the functions we call logarithms are interesting because they are the inverses of the exponential functions. But the development of logarithms and how they were used at the time had a much more practical purpose. They were needed for speedy calculations in situations such as navigation at sea. Indeed, jobs in surveying and accounting as well as calculating distances at sea, all required multiplication of this caliber on a frequent basis. In order to gain some appreciation of the role played by logarithms, we might try to compute the product of 6,560 and 980 without the aid of a calculator. It is not that difficult, but neither is it quick. As we will see, however, one can get a very accurate estimate of the answer quickly using logarithms. The method used for the 25 years prior to the introduction of logarithms ( ) was called prosthaphaeresis. This term which means addition (prosth-) and subtraction(-aphaeresis) uses laws of trigonometry to help find the product or quotient of two large numbers. Essentially, prosthaphaeresis allowed an astronomer to multiply two numbers as cosine values by averaging their sums and their differences. Instead of multiplying with pen and paper, a table of cosines aided in the following method of solution:

3 Suppose that the desired product is 147,809 * 6,756. The first step is to change both numbers into decimals (which is to say multiply by 10 raised to some negative integral exponent), a process known as scaling down. This yields and These values are to be treated as the cosine values of two angles. By looking up the two numbers in a table of cosines, one quickly determines that is the cosine of 81.5 degrees and (which rounds to.6756) is the cosine of 47.5 degrees. By using the trigonometric identity cos(a)cos(b) = [cos(a+b) + cos(a-b)]/2 which averages the sum and difference of the degree values (also known as the prosthaphaeretic rule) a rough decimal equivalent to the true value is obtained. That is to say: [cos( ) + cos( )]/2 is equal to cos(a)cos(b). By evaluating the computation inside of the parenthesis the statement is simplified to [cos(129) + cos(34)]/2 = cos(a)cos(b). Using a table of cosines, the previous statement is equivalent to [ ]/2 = cos(a)cos(b). The last computational step is to average the cosines found in the table; = cos(a)cos(b). This is not yet the true value of the product, but by scaling up the total number of places that had been initially scaled down, the correct placement of the decimal can be found. In the original problem 147,809 was changed into so that the number had been scaled down six places. Since 6756 changed into.6756, an additional four places had been scaled down. The scale down of 6 and 4 places sums to 10, hence the need to move the decimal ten places. The final step is to move the decimal in a total of ten place values to the right, producing the answer 998,600,000. This number is only an approximation of the true product, correct to 5 places, because the cosine values were only taken to 5 places. The prosthaphaeresis method was a great help because it substituted the use of table values together with addition and subtraction in place of the alternative method, which was to solve the problem through multiplication. Paul Wittich, Ibn Yunis, Joost Burgi, Johannes Werner Francois Viete and Christopher Clavius are all considered contributing participants to the

4 development of this approach to computation. In contrast, one man is credited with taking this information and manipulating it in order to be more accurate as well as more convenient. This enabled people to solve massive computation problems more efficiently. John Napier was born into wealth and nobility and spent a portion of his time assisting his father a tax collector. Napier found the process of multiplying and dividing large numbers tedious and set forth to do something about it. His initial goal was to speed up the process of the calculations (as prosthaphaeresis had done before him); and at the same time decrease the number of errors made while calculating. While there have been modifications since Napier s original thoughts on the subject were published in 1614 in Mirifici Logarithmorum Canonis Descriptio, the general concept put forth in the paper is still the same. Napier, however, was not the only mathematician/astronomer working on logarithms at the time. Within four years Joost Bürgi would publish his independent findings with the aid of Johannes Kepler. However, Bürgi and Kepler often take a backseat to Napier as the discoverer of logarithms since Napier s ideas were published first. John Napier s wealth and knowledge allowed him many privileges and the freedom to have many hobbies. Besides mathematics, he also delved into the areas of theology, astronomy, alchemy and necromancy (the summoning of spirits). While dabbling in mathematics, Napier viewed multiplication as being a shortcut for addition. Likewise, exponents were a shortcut for multiplication. Therefore there needed to be a shortcut for exponents. Napier used a common law of exponents, r m * r n = r (m+n) to aid him in finding a shortcut. By creating two lists of numbers, those that are absolute and those that are artificial, he discovered a way to turn multiplication problems into addition problems in a manner similar to prosthaphaeresis. Napier wanted to multiply large numbers (absolute numbers) quickly, so he substituted them with artificial numbers (logarithms) to turn the problem into one of addition instead of multiplication.

5 Napier started by using a large absolute number, 10 7, and matched it with an artificial number, 0. The number 10 7 was chosen for its ability to give seven figures of accuracy and because, more often than not, the numbers that needed multiplying by astronomers or navigators were smaller than By multiplying each absolute number by 1 1/10 7 or ( ) the next absolute number would be reached. According to Napier, to achieve the next artificial number, one simply adds one to the previous artificial number. The result will be two lists of numbers: one that is a geometric sequence and one that is an arithmetic sequence. For example, the first five absolute and artificial numbers are: Absolute Numbers Artificial Numbers 10,000, ,999, ,999, ,999, ,999,996 4 The absolute numbers are a geometric sequence due to the relation of multiplying the preceding term by a constant (in this case 1-1/10 7 ) in order to get the next term. The artificial numbers are an arithmetic sequence due to using addition of a constant to get to the next term. Decimals were not widely used during the era Napier was computing logarithms, so the value 10 7 was multiplied by each decimal to produce a whole number, giving the expression 10 7 (1-1/10 7 ) n. Initially Napier used the phrase artificial number to represent the exponent n; later he changed it to logarithm, a combination of the Greek terms: Logos (ratio) and arithmos (number). In essence the product of any two large numbers could be written as 10 7 (1-1/10 7 ) n * 10 7 (1-1/10 7 ) m, where n and m represent the logarithm of the number. Remembering the rule of exponents r n * r m = r (n + m) allows one to simply add n and m together to receive the logarithm of the product. This math procedure is allowed because the base for the exponents (m and n) is the same: namely, (1-1/10 7 ). To find the product of the original (absolute) numbers, one need only

6 look back through Napier s table for the term that the sum of n and m represents. The logarithmic value represented by the sum corresponds to the absolute number that is the product of the original two numbers. The concept of logarithms was of great interest to people such as Henry Briggs, a mathematics professor in London; when Briggs read the work of Napier, he was so enamored that by the summer of 1615, he went to Edinburgh to work with Napier to better develop logarithms. Before leaving for London, Briggs sent a letter (to Napier) detailing that if the use of base 10 would be possible, the calculations would be far more convenient. Napier had been working on the same idea but illness prevented him from creating such tables. Briggs purpose for using (what today would be called) a base of 10 would allow the evaluation of the logarithm of 1, which is equal to zero. The use of smaller values would prove most insightful when the attempt was made at making a visual representation of logarithms. Shortly after Napier died in 1617, Briggs published a book in which he gave a brief explanation of logarithms together with the logarithms of numbers from 1 to 1000, calculated to 14 decimal places. By 1624, Briggs published logarithms for the numbers from one to 20,000 and from 90,000 to 100,000 always using base 10. By 1628, he published the logarithms for 20,000 through 90,000. Log base 10, which is now called the common logarithm, helped make use of logarithmic properties of powers of 10, such as: log 1 = 0 log 10 = 1 log 100 = 2 This proved to be useful for other numbers as well such as: log 2.5 = log 25 = log 250 =

7 In all cases, it should be noted that when a number is increased by a factor of 10 the value of the logarithm of that number increases by a unit of 1. The logarithm values have two parts: the whole number, known as the characteristic and the decimal portion, called the mantissa. Using logarithms, an astronomer or a navigator no longer would need to complete multiplication or division problems by hand. A book of logarithmic values needed to be present in order to aid in the solving of the problem, however the amount of paper, ink, and time that could be saved was worth the small burden of carrying a book. Consider the problem presented in the introduction: What is 6560 multiplied by 980? Using the work of Napier and Briggs it can be determined that 6560 * 980 = 6,428,800. Here s how that may be confirmed using logarithms with a base of 10. First compute (by calculator or by looking it up in a table): Log (6560) = Log (980) = The next step is to add the values/answers = The final step involves working backwards, looking for the value in the table of values/answers. Once this number (or an approximation of this number) has been found, simply look to see what number has that logarithm and that s the product of the two original numbers. Later the term for looking backwards would be called finding the anti-logarithm, for the problem presented here that would be 6,428,800. Today, we would find the exact product using a calculator, whereas using Briggs table, which went out eight decimal places, exact answers could not always be retrieved. If the product of the two numbers needed more than eight significant digits, the logarithm method would only produce a value accurate to the first eight digits. It is at this point that some common sense must be used to determine how much precision is really necessary. For example, being accurate to

8 eight digits is about as accurate as computing the area of Nebraska to within one typical lot in Lincoln. It has been stated earlier that as the logarithmic number is increased by a power of ten, the part of its value known as the characteristic increases by one. This informs the problem solver where the decimal, if any, needs to go. For instance: Characteristic The value s largest place value 0 ones 1 tens 2 hundreds 3 thousands This was very important when the slide rule was invented around Thanks to the dedication of Briggs and the respect he commanded, there was in increase in the acceptance of logarithms and more and more people became involved with the advancement of logarithms. By 1620, the astronomer Edmund Gunter, decided to draw out the logarithms as a two-foot long number line in which the positions of the numbers were proportional to their logarithms. Gunter s scale started at 1, due to the fact that log(1)=0. The scale is as follows: Note that the distance between the numbers represents the proportional difference in their logs. As log(1) = 0 and log(10) = 1, this difference represents the total length of the stick. Since log(1)= 0 and log(2)=.30103, the difference of these two logarithms will be proportional to the length of the stick and must account for % of the total. Similarly, log(9) = and log(10)=1, two numbers which are close together. Hence the numbers 9 and 1 (the representative for 10) are positioned to provide that information. Note that to increase accuracy,

9 Gunter placed another logarithmically spaced scale in between 1 and 2. These numbers indicate logarithmic values between the 1 and 2. The more scales that could be put in between any two numbers, the more precise the multiplication could be. In order to multiply and divide with Gunter s line it was necessary to use calipers to measure the sum or differences of the logarithms. The findings of William Oughtred (to whom lead credit is generally given for the following findings) in 1622 would make the process of multiplying and dividing even easier. By taking two of Gunter s sticks and placing them side-by-side, the slide rule was created and the use of calipers became obsolete. To multiply two numbers using a slide rule, one first aligns the 1 (starting point) on the top scale with either of the two factors on the bottom scale. Then one moves along the top scale until the other factor is found. The number directly below that factor on the bottom scale is the product of the two numbers. For example, to multiply 18 by 2.2, align the top scale s starting point (1) with 1.8 (the number between 1 and 10 that corresponds to18) on the bottom scale. Next follow along on the top scale to where 2.2 is located. Directly below that is the product. Notice that it appears to be at or slightly before the 4 on the bottom scale. Only one challenge remains in using a slide rule. We would have done exactly the same thing if multiplying 18 times 22 or 18 times 2.2. Thus, there is a need to be able to figure out the location of the decimal point. For the given problem (18 * 2.2) we know that the answer is slightly more than 36, (2 multiplied by 18). Therefore, if the slide rule gave an answer of slightly less than four, it can be justified that the true solution for the product is a number that is slightly less than 4 multiplied by a power of 10 - in this case 40.

10 The following computation for 18 * 2.2 shows how the slide rule produces values similar to what might be obtained using logarithmic rules. log(18*2.2) = log(18) + log(2.2) = = Now take the antilogarithm of to get18*2.2 = , as was predicted with the slide rule. What is important here is that for the first time, the work of Napier and Briggs could be demonstrated by visual means. In the example, the connection between a slide rule and logarithms is seen in that the bottom scale is used to find the logarithm of 18, while the top scale is used to add the logarithm of the second factor (in this case 2.2) to the logarithm of the first factor. The sum of the two distances pinpoints the sum of these logarithms, thereby telling where the product of the two numbers would be on a log scale. If larger numbers are needed (as was the purpose of the invention) in the multiplication process, one considers the first 1 on the scale as some power of 10, then the final 1 on the scale represents the next power of 10. For example if one of the initial factors is 6,400, the first 1 would be a power of 10 as close as possible to but less than 6,400; in this case 1,000. The final 1 would be the next power of 10, namely 10,000. For an estimated 350 years, slide rules or slipsticks, as they are also called, were the means used to multiply and divide large numbers quickly. Through more development and scales on the slide rules, the user was also able to compute squares, square roots, cubes, cube roots, sines, cosines, tangents, and cotangents, among other things. The advent of the pocket calculator in the mid 1960 s and the scientific calculator around 1974 led to the demise of the slide rule because now people had an instrument capable of great computational speed and accuracy that could fit into their pocket. Thus, today s students are unfamiliar with the slide rule and how it works.

11 Michael Stifel introduced exponents in People within the realm of mathematics soon understood that 5 3 was a more convenient way of writing 5*5*5. The focus of exponential notation is the base; after all, the base is the number that is used in order to complete the multiplication problem. Briggs did not realize, or at least did not publish the idea, that by using the base of 10 when working with logarithms he was changing the emphasis of the base to an emphasis on the exponent. Certainly he and others understood that 10 3 = 1000, and that log(1000) = 3, but the connection between exponential and logarithmic form was not discovered until the late 1600 s. By using exponents, a subject taught in elementary school, to convey a connection and very important link to logarithms, a topic not taught until high school, the door was opened for advancements in logarithms. The common logarithm (base 10) was not the only type of logarithm that could be used; other bases are sometimes more helpful, depending on the context of the situation. Indeed logarithms might be described as the exponent to which the base needs to be raised to in order to produce a certain number. For example, Log 10 (1000) = 3 implies that 3 is the exponent to which the base (10) must be raised in order to produce the number This is the understanding of logarithms that is taught and used in today s mathematics. By 1865, John Wallis, an English mathematician who specialized in algebra, became the first person to write about logarithms as inverse relations to exponents in his piece known as Algebra. In 1694, Johann Bernoulli s work extended Wallis idea, resulting in the understanding and setup for logarithms that we use today. This work supported the thought that exponentials are simply numbers that emphasize the base, whereas logarithms are numbers that emphasize the exponent. Another way to think about this is to say that logarithm properties are just exponential properties expressed using language that focuses on the inverses of exponential functions.

12 Wallis and Bernoulli showed the ability to have different bases. One may ask, although Napier didn t say he was using a base, what base was he actually using? To find any of Napier s logarithms, the equation 10 7 (1-1/10 7 ) n was used. Ignoring the 10 7 prior to the parenthesis, (this number is not associated with the exponent, it is simply a constant being multiplied on in order to eliminate decimals) allows for examination of (1-1/10 7 ) n. If we think about applying this formula to larger and larger values of n, then at some point the n will be as large as 10 7, thereby changing the equation to (1-1/10 7 ) (10^7) or (1-1/n) n. It was not until 1727 that Leonhard Euler proved the limit of (1+1/n) n as n goes to infinity is the number now called e. Although this is slightly different from what Napier was using, there is a connection, namely that the base that Napier was using is 1/e, or more precisely, the base which Napier used approaches 1/e. Through slight modifications the use of base 1/e soon became just base e. What was originally described as log e (x) was soon written as ln(x) and referred to as the natural logarithm. The value e (also known as one of the natural constants ) is special, or natural, because it shows up in a lot of applications, for instance, at any time a quantity increases at a rate proportional to its value (exponential growth/decay). Today logarithms, and properties dealing with logarithms, have simply become (for some) something to memorize rather than understand due to time constraints in the school mathematics curriculum. These individuals miss the deep understanding that should go with a study of the concept. Looking at the history of the development of the concept, we see that Napier alone gave us two out of the three laws or distinct properties of logarithms. These laws are: 1. log b (xy) = log b (x) + log b (y) This property should look familiar. The basis for this property is found in Napier s work to simplify the multiplication of large numbers. The property says that finding the sum of the logarithms of two factors results in the logarithm of the product of the factors. 2. log b (x/y) = log b (x) log b (y) Similar to property one, it says that the difference of two logarithms will result in the logarithm of the quotient of the two numbers.

13 3. log b (x n ) = n log b (x) This property, if n is a positive integer, is actually just a modification on the first property. Since x n is x*x*x*x *x (n times) we may ascertain that log b (x n ) is log b (x) + log b (x) + log b (x)+ + log b (x) this would continue n times, or more simply arranged n * log b (x). Note that the three properties have their roots within the three laws of exponents: 1. b x * b y = b (x+y) 2. b x / b y = b (x-y) 3. (b (x)^y ) = b (x*y) Other facts worth noting are: Logarithmic rule Exponents rule Log b (1) = 0 b 0 = 1 Log b (b) = 1 b 1 = b Log b (b x )= x b x = b x This last one can be a little trickier than the rest if read aloud. What it is saying is: If b must be raised to some power so that it equals b x, then, clearly b must be raised to the power of x. Undoubtedly, when using logarithms, especially when using calculators, there are bases that are not easy to use. It is helpful to have a method to change bases so that a logarithm can be written in terms of a base that can easily be worked with. While mental computation can determine that log 7 (49) is equal to 2, what about log 7 (50)? It can be estimated to be slightly more than 2 but further precision is not readily accessible using mental calculations. There are, however, two ways to compute this value with a calculator even if the calculator only allows logarithms with bases of 10 and e: log b (x) = ln(x)/ln(b) or log b (x) = log(x)/log(b). The above are known as the change of base equations for logarithms. The following steps will explain how the change of base equations were derived.

14 First, start with the logarithmic equation, log b (x) = y. Then, rewrite the equation in terms of exponents, b y =x. The next move involves solving for y by taking the logarithm of both sides, but using a base other than b, log a (b y ) = log a (x). Since most calculators have a base 10 or an e key, a calculator can readily be used with one of these bases. To avoid deriving the problem for each base, the variable a will be used to cover both bases. Using logarithm property number 3, the left side of the equation may be simplified as y*log a (b) = log a (x). To get y by itself divide both sides by log a (b) so that y = log a (x) / log a (b). Initially log b (x) was equal to y, so it is necessary to again substitute the values in order to get back to our starting logarithm, log b (x) = log a (x) / log a (b). Note that this formula is valid for any choice of base a. For any positive value for a, log a (x) has a vertical asymptote at x = 0. The graph is open concave downwards and the graph crosses the x-axis at x = 1. Initially log a (x) grows more quickly from small positive values to when the graph intersects the x-axis at (1,0) then log a (x) grows more slowly. A logarithm graph is increasing for all positive numbers, thus there are no local maximums. One of the most important features of a logarithm function is that it grows at a very slow rate when x is larger than 1. If one considers the fact that each logarithm function is the inverse of an exponential function that grows very quickly, this is quite natural. Previously, we have hinted at, but not emphasized, the idea that exponentials and logarithmic functions are inverses of one another. From an algebraic point of view this is significant because it allows us to solve exponential problems where the variable is in the exponent, as seen in the following: 3 x = 2187 By taking the logarithm of both sides of this equation we see that log (3 x ) = log(2187). By logarithm property number 3, x * log(3) = log(2187) and dividing both sides by log(3), we obtain x = log(2187) / log(3) = 7. For this problem, we might have noticed that 3 7 = 2187, but not all math problems are answered as neatly as the problem above, hence the technique is valuable.

15 Similarly there are times when changing from a logarithm to an exponent makes solving a problem much easier. Here is an example: log(2x+3) = 3 By rewriting the logarithm as its inverse, an exponential, we see that 10 3 = (2x + 3), or 1000 = 2x + 3. It easily follows that x = 997/2. Finally, the graph of a logarithmic function and its inverse, an exponential function, is both useful and aesthetically pleasing. Note how the graph of one function can be obtained from the other function by reflecting its graph over the line y = x. As we stated earlier, the logarithm function grows very slowly as x gets larger, while its inverse, an exponential function grows very rapidly, as x gets larger. f(x) = 10^x g(x) = x h(x) = log_10(x) The above graph was created on Geogebra. This makes sense as inverse functions simply flip the values for x and y. For instance since ln(1) = 0 we plot (1,0) for the log function, and since its inverse is e x, e 0 = 1 and the point (0,1) is on the graph of the exponential function. Another phrase may be that the input of one function is the output for the inverse function.

16 It has been almost 400 years since Napier published his work that led to the development and understanding of logarithmic functions. Napier s work was motivated by the desire to find a better way to compute products efficiently and accurately, thus making life easier for himself and others. His discovery of logarithms had a tremendous influence on mathematics and society. Our job as teachers is to ensure that today s students understand this marvelous function and that it is not reduced to a button on a calculator or properties to memorize.

17 References: Ask Dr. Math (2008). High School: exponents/logarithms. Retrieved July 3, 2008, from: Berezovski, Tanya. (2006). Manifold Nature of Logarithms: number, operations and functions. All Academic Research. Retrieved July 2, 2008, from Better Explained: Learn Right, Not Rote. (2007). Demystifying the Natural Logarithm (ln). Retrieved July 1, 2008, from Dawkins, Paul. (2008). Paul s Online Math Notes. Retrieved July 2, 2008, from Lamar University, Department of Mathematics Web site: Eric s Slide Rule Sight. (2001). Introduction to the Slide Rule. Retrieved June 30, 2008, from The History of Computing Project. (2008). John Napier. Retrieved July 2, 2008, from Interactive Mathematics. (2008). Exponential and Logarithmic Functions. Retrieved

18 June 30, 2008, from functions/exponential-log-functions-intro.php Logarithms. (2004). In Hutchinson Encyclopedia. Retrieved June 30, 2008, Hutchinson Encyclopedia Online: The Math Page. (2008). Topics in Precalculus. Retrieved July 1, 2008, from: The Old Computer Hut. (2002). Napier s Logarithms. Retrieved July 2, 2008, from Otero, Daniel. (2000). Henry Briggs: Need for Fast Calculation. Retrieved July 1, 2008, from Xavier University, Department of Computer Web site: The Oughtred Society. (2007). Slide Rule History. Retrieved July 1, 2008, from S.O.S. Mathematics. (2008). A Review of Logarithms. Retrieved July 1, 2008, from