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Logarithms

Overview

A logarithm is the power to which a number (usually termed the base number) must be raised to equal a target number.

Fundamental Mathematical Concepts and Terms

In base 10 systems, 2 is the logarithm of 100 because 10² = 10 × 10 = 100. The number 2 in this example is the exponent of the base number 2 that yields 100. Accordingly, in base 10 the log of 100 is 2.

Because logarithms are so common in mathematics, there are different ways to develop an understanding of them and this will often cause some confusion. However, at their most basic level they can be thought of as a set of rules that allow one quantity to be converted into another to simplify a problem. This idea is the same as multiplication, which is a simplification of the operation of repeated addition; it is easier to say 50 × 5 rather than write 50 + 50 + 50 + 50 + 50. Logarithms are effectively the next step, the simplification of repeated multiplication or division. Logarithms have their own form of mathematical notation that can only be manipulated in strict accordance with a set of rules. Once these rules are understood the work of manipulating logarithms is carried by the notation itself, operations no more complex than multiplication, addition, subtraction and division are used to manipulate terms in an equation and generate the desired result.

Before trying to use the mathematical notation of logarithms, it is helpful to understand some aspects of mathematical notation itself. Consider the simple operation of multiplication. The common notation for multiplication is the × symbol.

Using a symbol to stand for repeated addition, called multiplication, is a form of shorthand. For example, 2 × 3 = 2 + 2 + 2. Obviously, the notation reduced the amount of work needed to express repeated addition; it would be hard work if you had to write out 3,200 × 563 as 3,200 + 3,200 + 3,200 … some five hundred sixty three times. However, this notation is more than just a shorthand; it allows us to manipulate quantities that were not possible before and extend our range of mathematical tools. For example, consider multiplying two fractions together. Even though it is not possible to write this out directly, it is still possible to find the answer, 0.5 × 0.7 = .35. It is even possible to throw away the numbers and replace them with letters that represent any number that you can think of. Here x means any number we can think of multiplied by three. y now represents the answer, 3 × x = y. In equations like this the multiplication symbol, ×, is often dropped and letters and numbers that are next to each other are understood to be multiplied. If we set x = 2, remembering this is just a number used for example, our equation y is found to be equal to 3 × 2 = 6. Again, if x = 5, then y is equal to 15.

One of the great powers of this notation is that it allows the terms, such as x and y, to change places. This is done by noticing that any operation performed to on one side of the equal sign must be repeated on the other side of the equal sign. This is because the values are equal and what we do to one side should balance the effect on the other side.

Let's consider an example of this for the formula 3 × x = y. This formula may give us some property of a material, and we conduct an experiment where we measured the values of y of that material. Can we find the value of x? Yes, simply find the value of x by dividing both sides by three: (3 × x)/3 = y/3 so this gives x = y/3. By dividing the equation by 3 on both sides we have eliminated the term in front of the x and given the equation in terms of y. This process is called rearranging an equation.

THE POWER OF MATHEMATICAL NOTATION

As multiplication was the extension of repeated addition, so raising to powers, or simply powers, is the extension of repeated multiplication. Consider the number 5 multiplied by itself four times; this can be written in shorthand by putting the number of times the multiplication is to be repeated as a smaller number 4 to the top right of the 5. Here are some examples, 5 × 5 × 5 × 5 = 5⁴ is the same as 5 × 5 × 5 × 5 = 5⁴. To read this notation out loud we say base five raised to the power of four. Another example is 2 × 2 × 2 × 2 × 2 × 2 = 2⁶ read as base two raised to the power of six. There are a couple of points to remember about this notation. The first is that the power is also known as the exponent and raising to a power is also known as exponentiation. Exponentiation is not to be confused with the exponential function, e^x, discussed later. Another point to note is that numbers raised to the power of two or three are often read as squared or cubed. For example, 5 × 5 = 5² is read as five squared and 8 × 8 × 8 = 8³ is read as eight cubed.

As division is the opposite of multiplication, logarithms can be thought of as the opposite of exponents. They come in two common forms. The first form is written as log₁₀, read as log base 10. The base here is related to the base of the powers, as we shall soon see. Log₁₀ is so common that in texts and the buttons on most calculators the base 10 is dropped and it simply reads as log or lg. The other form is read as the natural logarithm, and is written as ln, this is identical to "log e." Logs to any other base, such a base 2, are written as log₂, etc.

POWERS AND LOGS OF BASE 10

Now students should try to get a feel for some values in base 10. Using a scientific calculator, they can try the following, log (1,000) = 3. This tells us that 1,000 can be repeatedly divided by 10 three times: 1,000 /10 /10 /10 = 1, which is true. Another way to look at this is the logarithm has told us that the number 1000 has three zeros after the one. Now raise base 10 to the power of 3 and we are back with the number we started from, 10 × 10 × 10 = 10³ = 1,000. Here we see the relation between the base of the logarithm and the base of the power. This reflects the relationship of logarithms as repeated division and powers as repeated multiplication. Raising the logarithm to the power like this is called an anti-logarithm, and it gives us back the number we started with. Here is another example, log (10,000) = 4. Again, this shows us that 10,000 can be repeatedly divided by 10 exactly four times, or to view it another way, there are four zeros after the 1. Raising this logarithm to the power of base 10 gives us back our number, 10⁴ = 10,000.

For any number made from one followed by a number of zeros the log will always equal the number of zeros if we use logarithms with base 10.

As with the previous multiplication example, our definitions of this notation allow us to extend this idea of repeated multiplication and division to more than just shorthand, because we can now use fractional values. Students should try the following, log (5,246) = 3.7198283. Even though this cannot be written out as an exact repeated division by ten as we did before, it still tells us how 5,246 would divide into 10 in an abstract sense, about four times. If we raise this to base ten do we get the answer back as expected? 10^3.7198283 = 5,246.0002. Almost, but what about the small fraction after the number? (Depending on places and the calculator, the exact fraction may differ.) The digits after the decimal place are not important and are there because the calculator cannot store numbers to infinite precision. However, they can safely be ignored as the error is not in the digits in which we are interested. This will always be found to be true and we obtain the correct answer of 5,246. The notation has allowed the extension of the mathematical idea of repeated multiplication to be taken beyond the simple idea of a shorthand.

LOGARITHMS TO OTHER BASES THAN 10

What about logarithms with a base other than 10, such as, log₂ (256) = 8? You do not find a log₂ button on your calculator because logarithms to bases other than 10 can always be expressed as log₁₀ using the following formula:
logN (y) = log₁₀ (y)/log₁₀ (N)

Here y is the value of the log and N is the value of the base. So, to solve the previous equation, log₂ (256) = log₁₀ (256)/log₁₀ (2) = 2.40824/0.30103 = 8. As a check, 2⁸ = 256, as expected.

Logarithms to the base 2 are common in computing where a computer will represent numbers by a series of 1s or 0s internally. Arithmetic performed in base two is called binary.

POWERS AND THEIR RELATION TO LOGARITHMS

Let us consider replacing the numbers with letters as was done with multiplication. Again the letter x can take any value, y = 10^x. If we apply log to the terms on both sides of the equals sign we can now find x. Check the method used for rearranging the formula, 3x = y, if you do not understand this step, log (y) = log (10^x) = x x = log (y). This shows the effect of the log was to cancel the base of the power, 10, in y = 10^x. This is the same in any base and generally can be written log_N (N^x) = x. Notice that the base must be the same in both parts. For instance, the following formula is wrong, log₂ (3^x) is not equal to x but this is correct, log₃ (3^x) = x. This rule allows us to cancel the base of a power by multiplication with a logarithm. This is useful for extracting the power x.

THE ALGEBRA OF POWERS AND LOGARITHMS

When two powers of the same base are multiplied, the repeated multiplication is effectively extended. This is identical to the base raised to the sum of the powers. If they are divided, then the powers are just subtracted. N^x × N^y = N^(x+y) (N^x)/(N^y) = N^(x−y).

While thinking about these relations, students can see that we have combined multiplication and addition and vise-versa for division and subtraction. Using the logarithm to extract the powers, shown previously, allows the addition and subtraction parts to be extracted, Log_N (N^x × N^y) = x + y = log_N (N^x) + log_N (N^y) and if we set N^x = A and N^y = B then log_N (A × B) = log_N (A) + log_N (B). Likewise log_N (A/B) = log_N (A) − log_N (B). The rules shown here are the reason that logarithms allow us to reduce the complexity of large lists of multiplications (or divisions) down to simple addition (or subtraction).

These basic properties of logarithms were critical in the development of science and industry over the past three centuries.

LOG TABLES

Suppose you want to multiply numbers so large that it will take a while to complete the computation by hand. It would be faster if there were some sort of table to look up the answer. If we wanted the answer to be accurate to four digits, a simple solution would be to make a table, called a matrix, with the rows and columns corresponding to the numbers between 1 and 9,999. If we picked a row, say 50 and column say 26, were they crossed we would find the answer for the multiplication, in this case 1300. Picking a row and column would show us the multiplied answer quickly.

Now any multiplication can simply be looked up in our table. For numbers outside the range 1 to 9,999 we can still find the answer by moving the decimal point until they are in this range, reading from the table and finally moving the decimal place back by an opposite number of steps.

The problem with this basic system, that makes it unworkable, is the number of entries needed will be huge. For four digits accuracy each edge of our square table would have 10,000 numbers. This gives us least 100,000,000 entries (10,000²). If the print is very small that is still enough to fill 20 thick books.

Another problem is seen as we increase the number of digits accuracy needed. The square shape of our table, the matrix, will rapidly start to get bigger with each digit added. Most scientific and engineering calculations work at seven digits accuracy. This works out to be more than a million thick books to store our table and we have not even considered division.

History of Logarithms

Logarithms were invented in the seventeenth century by John Napier, a Scottish Barron. During that time in Scottish history the country was undergoing major religious and political upheaval. In this climate academic study was not held in high regard. Later in life Napier considered his greatest publications were his theological works, with his mathematical works as a secondary interest. The development of logarithms at this time came from a need to simplify the computations of repeated multiplications and divisions. These computations were common in the calculations of astronomical charts used by the navy and the shipping industry and religious charts used by the church, three of the most powerful institutions in Britain at the time.

The power of the system of logarithms comes from its simplification of computational steps involved. By converting terms that were to be multiplied or divided to logarithms from a table, they could then simply be added or subtracted, and the result read from another table. The system was compact and flexible enough that the two tables needed to perform the steps could be listed in a few pages at the back of a book. Another device, called a slide rule, reduced computational time further by allowing the user to read the answer directly by simply moving the two rulers on the device. For 300 years this device was commonly used by scientists and engineers, just as hand-held calculators are today.

It is no understatement that the invention of logarithms changed the world. Their usefulness in industry and science was soon realized, and the system rapidly spread around the world. The invention of the electronic calculator in the 1950s allowed complex calculations to be performed by the simple push of a button.

Real-life Applications

COMPUTER INTENSIVE APPLICATIONS

Although the system of using log tables is not in common use since the invention of electronic calculators it has found new life in computer intensive applications. The desktop calculator will have to run a program to multiply numbers together. This process is so fast we cannot see it and to us it looks instant. However, it still takes a certain amount of time, and the time needed increases with the complexity and amount of calculations. In a computer-intensive application, in which millions of numbers have to be multiplied every second and speed is critical, this can become a problem. Some examples are interactive 3D computer games, and software used in spacecraft and aircraft. Here, the small time the computer takes to calculate the numbers will rapidly increase and can become substantial.

This can be reduced if a set of log and anti-log tables are wired into the computers memory, and the computer only has to look up values instead of running a program to find the answer. This technique to improve performance under heavy arithmetic load is called a log lookup table.

USING A LOGARITHMIC SCALE TO MEASURE SOUND INTENSITY

Decibels (dB) are used as a measure of sound level. They are common markings for stereos, televisions, and other audio equipment and are based on a log10 scale. The faintest sound we can hear is called the threshold of hearing. Its value is tiny, about a 0.3 billionths change in air pressure. The scale is given as dB = log (Number of times greater than threshold of hearing) × 10. A normal conversation is 60dB or, remembering to divide by the 10 from right of the formula, 10⁶ = 1,000,000. This is one million times louder than the faintest sound you can hear. We can safely hear sounds to around 90 dB, the level of an orchestra, before damage to the ear starts to occur, but we can still hear sounds louder than that. The levels of the front row of a rock concert can reach 110 dB. After this, there is pain and instant deafness. This gives the human ear an amazing range of about 100 billion times the faintest sound it can detect.

ESTIMATING THE AGE OF ORGANIC MATTER USING CARBON DATING

The atmosphere is continuously being bombarded by radiation from space. In the upper atmosphere, the radiation from space has enough energy to change atoms of nitrogen into carbon. Carbon created this way is called carbon 14 and is different to the majority of the carbon we see around us called carbon 12. Carbon 14, unlike carbon 12, is unstable and will slowly decay back to nitrogen over a period of many thousands of years. The rate of production of carbon 14 in the atmosphere can be shown to be stable for a very long period of history, and this allows us to measure the age of dead organic matter.

All life on Earth is made from carbon, and during the course of an organism's life it will absorb small amounts of carbon 14. When the organism dies, it will stop absorbing carbon 14. So by measuring the ratio of carbon 14 to carbon 12 that is present and using the law of exponential decay of a radioactive source and their logarithms, scientists can calculate the age of the material.

DEVELOPING OPTICAL EQUIPMENT

No matter how pure a material is made, as light passes through it a small majority will always be scattered or absorbed. This is an exponential effect and logarithms are therefore used extensively in the design of optical equipment. Just a few examples are cameras, optical fibers, and the design of television screens.

USE IN MEDICAL EQUIPMENT

Certain cancers can be treated by passing radioactive beams though the body. A machine with a radioactive source is rotated around the patient. Only at the center of this rotation will the radiation be constant. Moving away from the center, the radiation will only pass through the patient periodically as the machine makes each rotation.

The location of the cancer is carefully mapped, and the absorption of the radiation through the body and the absorption by various organs is then calculated. This requires the use of logarithms due to the exponential nature of this absorption. The aim of the surgeons is to locate the cancer and manipulate the intensity of the beam over each rotation so that minimum damage is caused to the surrounding organs and maximum damage is caused to the tumor.

DESIGNING RADIOACTIVE SHIELDING FOR EQUIPMENT IN SPACE

Outside the protection of the Earth's atmosphere we enter a highly radioactive environment. Spaceships, satellites, and spacesuits must all able to absorb and disperse this energy to protect the delicate equipment and astronauts from damage. The absorption must be balanced against weight not too massive to launch.

Absorption of different types of radiation and in different materials involves calculations using logarithms due to the exponential nature of absorption.

SUPERSONIC AND HYPERSONIC FLIGHT

During supersonic and hypersonic flight, the air flow over the craft behaves very differently than at slower flight speeds. Logarithms are used in the design and fuel requirements.

Potential Applications

CRYPTOGRAPHY AND GROUP THEORY

Cryptography is the science of encoding information in such a way that an eavesdropper cannot intercept and decode a message. Modern methods rely on a mathematical phenomenon that some formulas are practically impossible to invert. This means that information encoded by such a formula cannot simply be decoded by rearranging the terms of the formula to reverse the processes.

Two parties can generate and swap unique keys which will unlock the message encrypted by a formula like this. However, if an eavesdropper were to try to decode the encryption by setting a machine up between the two parties without the keys, he would have to invert the formula used to encrypt the information.

One set of such functions that show these properties come from an abstract area of mathematical research that studies the relations between objects called group theory. Certain groups can be given properties that act like exponentials and logarithms. The calculation of the exponential part of these groups is very simple, and the calculation of the logarithmic part is very hard. This property can be exploited in cryptography. Studies of this branch of mathematics are important in the future development of faster and more secure algorithms.

Where to Learn More

Books

Durbin, John R. College Algebra. New York: John Wiley & Sons, 1985.

Morrison, Philip and Phylis Morrison. Powers of Ten: A Book About the Relative Size of Things in the Universe and the Effect of Adding Another Zero. San Francisco: Scientific American Library, 1982.

Periodicals

Curtis, Lorenzo. "Concept of the exponential law prior to 1900." American Journal of Physics 46(9), Sep. 1978, pp. 896–906 (also available at <http://www.physics.utoledo.edu/~ljc/explaw.pdf>.

Web sites

SOS Math! "Introduction to logarithms." <http://www.sosmath.com/algebra/logs/log1/log1.html> (February 1, 2005).