Logarithms

Resources

Finding the logarithm of a number is the inverse of raising the number to an exponent (exponentiation). In general, the base b logarithm of any number x is the number L such that x = b^L. For example, the base 10 logarithm of 100 is 2 because 100 = 10². This can be abbreviated log₁₀100 = 2.

Because logarithms are defined in terms of exponents, they have an intimate connection with exponential functions and with the laws of exponents.

The basic relationship is b^x = y if and only if x = log_b y. Because 2³ = 8, log₂ 8 = 3. Because, according to a table of logarithms, log₁₀ 2 = 0.301, 10^.301 =2.

The major laws of logarithms and the exponential laws from which they are derived are shown in Table 1.

In all these rules, the bases a and b and the arguments x and y are limited to positive numbers. The exponents m, n, p, and r and the logarithms can be positive, negative, or zero.

Because logarithms depend on the base that is being used, the base must be clearly identified. It is usually shown as a subscript. There are two exceptions. When the base is 10, the logarithm can be written without a subscript. Thus log 1000 means log₁₀ 1000. Logarithms with 10 as a base are called “common” or “Briggsian.” The other exception is when the base is the number e (which equals 2.718282. . .). Such logarithms are written ln x and are called “natural” or “Napierian” logarithms.

In order to use logarithms one must be able to evaluate them. The simplest way to do this is to use a “scientific” calculator. Such a calculator will ordinarily have two keys, one marked “LOG,” which will give the common logarithm of the entered number, and the other “LN,” which will give the natural logarithm.

Lacking such a calculator, one can turn to the tables of common logarithms found in various handbooks or appendices of various statistical and mathematical texts. In using such tables one must know that they contain logarithms in the range 0 to 1 only. These are the logarithms of numbers in the range 1 to 10. If one is seeking the logarithm of a number, say 112 or 0.0035, outside that range, some accommodation must be made.

The easiest way to do this is to write the number in scientific notation:

112 = 1.12 × 10²
.0035 = 3.5 × 10^-3

Then, using law I

log 112 = log 1.12 + log 10²
log .0035 = log 3.5 + log 10^-3

Log 1.12 and log 3.5 can be found in the table. They are 0.0492 and 0.5441 respectively. Log 10² and log 10^-3 are simply 2 and -3 according to law V: therefore

log 112 = .0492 + 2 = 2.0492
log .0035 = .5442 - 3 = -2.4559

The two parts of the resulting logarithms are called the “mantissa” and the “characteristic.” The mantissa is the decimal part, and the characteristic, the integral part. Since tables of logarithms show positive mantissas only, a logarithm such as -5.8111 must be converted to 0.1889-6 before a table can be used to find the “antilogarithm,” which is the name given to the number whose logarithm it is. A calculator will show the antilogarithm without such a conversion.

Tables for natural logarithms also exist. Since for natural logarithms, there is no easy way of determining the characteristic, the table will show both characteristic and mantissa. It will also cover a greater range of numbers, perhaps 0 to 1000 or more. An alternative is a table of common logarithms, converting them to natural logarithms with the formula (from law IV) ln x = 2.30285× log x. Logarithms are used for a variety of purposes. One significant use—the use for which they were first invented—is to simplify calculations. Laws I and II enable one to multiply or divide numbers by adding or subtracting their logarithms. When numbers have a large number of digits, adding or subtracting is usually easier. Law III enables one to raise a number to a power by multiplying its logarithm. This is a much simpler operation than doing the exponentiation, especially if the exponent is not 0, 1, or 2.

At one time logarithms were widely used for computation. Astronomers relied on them for the extensive computations their work requires. Engineers did a majority of their computations with slide rules, which are mechanical devices for adding and subtracting logarithms or, using log-log scales, for multiplying them. Modern electronic calculators have displaced slide rules and tables for computational purposes—they are quicker and far more precise—but an understanding of the properties of logarithms remains a valuable tool for anyone who uses numbers extensively.

If one draws a scale on which logarithms go up by uniform steps, the antilogarithms will crowd closer and closer together as their size increases. This is done in a very systematic way. On a logarithmic scale, as this is called, equal intervals correspond to equal ratios. The interval between 1 and 2, for example, is the same length as the interval between 4 and 8.

Logarithmic scales are used for many purposes. The pH scale used to measure acidity and the decibel scale used to measure loudness are both logarithmic scales (that is, they are the logarithms of the acidity and loudness). As such, they stretch out the scale where the acidity or loudness is weak (and small variations noticeable) and compress it where it is strong (where big variations are needed for a noticeable effect). Another example of the advantage of a logarithmic scale can be seen in a scale that a sociologist might construct. If he were to draw an ordinary graph of family incomes, an increase of a dollar an hour in the minimum wage would seem to be of the same importance as a dollar-an-hour increase in the income of a corporation executive earning a half million dollars a year. Yet such an increase would be of far greater importance to the family whose earner or earners were working at the minimum-wage level. A logarithmic scale, where equal intervals reflect equal ratios rather than equal differences, would show this.

Logarithmic functions also show up as the inverses of exponential functions. If P = ke^t, where k is a constant, represents population as a function of time, then t = k + ln P, where K = -ln k, is also a constant, represents time as a function of population. A demographer wanting to know how long it would take for the population to grow to a certain size would find the logarithmic form of the relationship the more useful one.

Because of this relationship logarithms are also used to solve exponential equations, such as 3 - = 2x as or 4e k = 15.

The invention of logarithms is attributed to John Napier, a Scottish mathematician who lived from 1550 to 1617. The logarithms he invented, however, were not the simple logarithms we use today (his logarithms were not what are now called “Napierian”). Shortly after Napier published his work, Briggs, an English mathematician met with him and together they worked out logarithms that much more closely resemble the common logarithms that we use today. Neither Napier nor Briggs related logarithms to exponents, however. They were invented before exponents were in use.

KEY TERMS

Characteristic —The integral part of a logarithm.

Logarithm —An exponent. If a = b^c, c is the logarithm to the base b of a.

Logarithmic function —A function of the form y = K + log_b x.

Logarithmic scale —A scale in which the logarithms of the numbers are uniformly spaced.

Mantissa —The decimal part of a logarithm.

Resources

BOOKS

Bicher, Maxime and Gaylord, Harry Davis. Trigonometry with Theory and Use of Logarithms. Scholarly Publishing Office, University of Michigan Library, 2005.

Lial, Margaret L., et al. Precalculus. Indianapolis, IL: Addison Wesley, 2004.

Stewart, James, et al. Precalculus: Mathematics for Calculus. Belmont, CA: Brooks Cole, 2005.

J. Paul Moulton

Logarithms

A logarithm is an exponent . The logarithm (to the base 10) of 100 is 2 because 102 = 100. This can be abbreviated log10100 = 2.

Because logarithms are exponents, they have an intimate connection with exponential functions and with the laws of exponents.

The basic relationship is bx = y if and only if x = logb y. Because 23 = 8, log2 8 = 3. Because, according to a table of logarithms, log10 2 =0.301, 10.301 = 2.

The major laws of logarithms and the exponential laws from which they are derived are shown in Table 1.

In all these rules, the bases a and b and the arguments x and y are limited to positive numbers. The exponents m, n, p, and r and the logarithms can be positive, negative , or zero .

Because logarithms depend on the base that is being used, the base must be clearly identified. It is usually

shown as a subscript. There are two exceptions. When the base is 10, the logarithm can be written without a subscript. Thus log 1000 means log10 1000. Logarithms with 10 as a base are called "common" or "Briggsian." The other exception is when the base is the number e (which equals 2.718282...). Such logarithms are written ln x and are called "natural" or "Napierian" logarithms.

In order to use logarithms one must be able to evaluate them. The simplest way to do this is to use a "scientific" calculator . Such a calculator will ordinarily have two keys, one marked "LOG," which will give the common logarithm of the entered number, and the other "LN," which will give the natural logarithm.

Lacking such a calculator, one can turn to the tables of common logarithms found in various handbooks or appendices of various statistical and mathematical texts. In using such tables one must know that they contain logarithms in the range 0 to 1 only. These are the logarithms of numbers in the range 1 to 10. If one is seeking the logarithm of a number, say 112 or 0.0035, outside that range, some accommodation must be made.

The easiest way to do this is to write the number in scientific notation:

Then, using law I

Log 1.12 and log 3.5 can be found in the table. They are 0.0492 and 0.5441 respectively. Log 102 and log 10-3 are simply 2 and -3 according to law V: therefore

The two parts of the resulting logarithms are called the "mantissa" and the "characteristic." The mantissa is the decimal part, and the characteristic, the integral part. Since tables of logarithms show positive mantissas only, a logarithm such as -5.8111 must be converted to 0.1889-6 before a table can be used to find the "antilogarithm," which is the name given to the number whose logarithm it is. A calculator will show the antilogarithm without such a conversion.

Tables for natural logarithms also exist. Since for natural logarithms, there is no easy way of determining the characteristic, the table will show both characteristic and mantissa. It will also cover a greater range of numbers, perhaps 0 to 1000 or more. An alternative is a table of common logarithms, converting them to natural logarithms with the formula (from law IV) ln x = 2.30285 × log x. Logarithms are used for a variety of purposes. One significant use—the use for which they were first invented—is to simplify calculations. Laws I and II enable one to multiply or divide numbers by adding or subtracting their logarithms. When numbers have a large number of digits, adding or subtracting is usually easier. Law III enables one to raise a number to a power by multiplying its logarithm. This is a much simpler operation than doing the exponentiation, especially if the exponent is not 0, 1, or 2.

At one time logarithms were widely used for computation. Astronomers relied on them for the extensive computations their work requires. Engineers did a majority of their computations with slide rules, which are mechanical devices for adding and subtracting logarithms or, using log-log scales, for multiplying them. Modern electronic calculators have displaced slide rules and tables for computational purposes—they are quicker and far more precise—but an understanding of the properties of logarithms remains a valuable tool for anyone who uses numbers extensively.

If one draws a scale on which logarithms go up by uniform steps, the antilogarithms will crowd closer and closer together as their size increases. They do this in a very systematic way. On a logarithmic scale, as this is called, equal intervals correspond to equal ratios. The interval between 1 and 2, for example, is the same length as the interval between 4 and 8.

Logarithmic scales are used for many purposes. The pH scale used to measure acidity and the decibel scale used to measure loudness are both logarithmic scales (that is, they are the logarithms of the acidity and loudness).


As such, they stretch out the scale where the acidity or loudness is weak (and small variations noticeable) and compress it where it is strong (where big variations are needed for a noticeable effect). Another example of the advantage of a logarithmic scale can be seen in a scale which a sociologist might construct. If he were to draw an ordinary graph of family incomes, an increase of a dollar an hour in the minimum wage would seem to be of the same importance as a dollar-an-hour increase in the income of a corporation executive earning a half million dollars a year. Yet such an increase would be of far greater importance to the family whose earner or earners were working at the minimum-wage level. A logarithmic scale, where equal intervals reflect equal ratios rather than equal differences, would show this.

Logarithmic functions also show up as the inverses of exponential functions. If P = ket, where k is a constant, represents population as a function of time, then t = K + ln P, where K = -ln k, is also a constant, represents time as a function of population. A demographer wanting to know how long it would take for the population to grow to a certain size would find the logarithmic form of the relationship the more useful one.

Because of this relationship logarithms are also used to solve exponental equations, such as 3 - = 2x as or 4e k = 15.

The invention of logarithms is attributed to John Napier, a Scottish mathematician who lived from 1550 to 1617. The logarithms he invented, however, were not the simple logarithms we use today (his logarithms were not what are now called "Napierian"). Shortly after Napier published his work, Briggs, an English mathematician met with him and together they worked out logarithms that much more closely resemble the common logarithms that we use today. Neither Napier nor Briggs related logarithms to exponents, however. They were invented before exponents were in use.

Resources

books

Finney, Thomas, Demana, and Waits. Calculus: Graphical, Numerical, Algebraic. Reading, MA: Addison Wesley Publishing Co., 1994.

Gullberg, Jan, and Peter Hilton. Mathematics: From the Birth of Numbers. W.W. Norton & Company, 1997.

Hodgman, Charles D., ed. C.R.C. Standard Mathematical Tables. Cleveland: Chemical Rubber Publishing Co, 1959.

Turnbull, Herbert Westren. "The Great Mathematicians." in The World of Mathematics. Edited by James R. Newman. New York: Simon and Schuster, 1956.

J. Paul Moulton

KEY TERMS

Conic section

—A conic section is a figure that results from the intersection of a right circular cone with a plane. The conic sections are the circle, ellipse, parabola, and hyperbola.

Line

—A line is a collection of points. A line has length, but no width or thickness.

Plane

—A plane is also a collection of points. It has length and width, but no thickness.

Point

—In geometric terms a point is a location. It has no size associated with it, no length, width, or thickness.

Right circular cone

—The surface that results from rotating two intersecting lines in a circle about an axis that is at a right angle to the circle of rotation.

Logarithms

The logarithm of a positive real number x to the base-a is the number y that satisfies the equation a ^y = x. In symbols, the logarithm of x to the base-a is log_a x, and, if a ^y = x, then y = log_a x.

Essentially, the logarithm to base-a is a function: To each positive real number x, the logarithm to base-a assigns x a number y such that a ^y = x. For example, 10² = 100; therefore, log₁₀ 100 = 2. The logarithm of 100 to base-10 is 2, which is an elaborate name for the power of 10 that equals 100.

Any positive real number except 1 can be used as the base. However, the two most useful integer bases are 10 and 2. Base-2 , also known as the binary system, is used in computer science because nearly all computers and calculators use base-2 for their internal calculations. Logarithms to the base-10 are called common logarithms. If the base is not specified, then base-10 is assumed, in which case the notation is simplified to log x.

Some examples of logarithms follow.

log 1 = 0 because 10⁰ = 1

log 10 = 1 because 10¹ = 10

log 100 = 2 because 10² = 100

log₂ 8 = 3 because 2³ = 8

log₂ 2 = 1 because 2¹ = 2

log₅ 25 = 2 because 5² = 25

log₃ = −2 because 3^-2 =

The logarithm of multiples of 10 follows a simple pattern: logarithm of 1,000, 10,000, and so on to base-10 are 3, 4 and so on. Also, the logarithm of a number a to base-a is always 1; that is, log_a a = 1 because a ¹ = a.

Logarithms have some interesting and useful properties. Let x, y, and a be positive real numbers, with a not equal to 1. The following are five useful properties of logarithms.

1. log_a(xy ) = log_a x + log_ay, so log₁₀(15) = log₁₀ 3 + log₁₀ 5

2. log_a = log_a x − log_a y, so log (⅔) = log 2 − log 3

3. log_a x ^r = r log_a x, where r is any real number, so log 3⁵ = 5 log 3

4. log_a = −log_a x, so log (¼) = (−1) log 4 because ¼ = (4)^-1

5. log_a a ^r = r, so log₁₀ 10³ = 3

Logarithms are useful in simplifying tedious calculations because of these properties.

History of Logarithms

The beginning of logarithms is usually attributed to John Napier (1550–1617), a Scottish amateur mathematician. Napier's interest in astronomy required him to do tedious calculations. With the use of logarithms, he developed ideas that shortened the time to do long and complex calculations. However, his approach to logarithms was different from the form used today.

Fortunately, a London professor, Henry Briggs (1561–1630) became interested in the logarithm tables prepared by Napier. Briggs traveled to Scotland to visit Napier and discuss his approach. They worked together to make improvements such as introducing base-10 logarithms. Later, Briggs developed a table of logarithms that remained in common use until the advent of calculators and computers. Common logarithms are occasionally also called Briggsian logarithms.

see also Powers and Exponents.

Rafiq Ladhani

Bibliography

James, Robert C., and Glenn James. Mathematics Dictionary, 5th ed. New York: Van Nostrand Reinhold, 1992.

Young, Robyn V., ed. Notable Mathematicians, from Ancient Times to the Present. Detroit: Gale Research, 1998.

Logarithm

In the 1500s and early 1600s, although science, engineering, and medicine were flourishing, many people did not understand multiplication tables. Mathematicians, astronomers, navigators, and scientists were forced to spend a lot of time performing calculations, so that little time was left to work on experiments and new discoveries. Finally, around 1594 Scottish mathematician John Napier (1550–1617) produced a table of logarithmic, or proportionate, numbers.

How logarithms work

In the commonly known base 10 system, computations that involve very large numbers can become difficult, if not incomprehensible. Napier realized numbers could be more easily expressed in terms of powers. Thus 100 is equal to 10 multiplied by 10, written as 10². This is read as "10 squared" and means "10 to the power two."

To perform multiplication, numbers are converted into logarithms, the exponents added together, and the result converted back into base 10. Likewise, to perform division, two logarithmic exponents are subtracted, and the result converted back to base 10.

This innovative way of multiplying and dividing large numbers was a milestone event for mathematicians of the day. Napier's tables were published in 1614 and were put into use immediately, becoming an essential part of the mathematical, scientific, and navigational processes.

Logarithmic tables remained popular throughout the next several centuries and were used as the basis for many mechanical calculating devices. Relieved from much of their mental drudgery, scientists and mathematicians enjoyed new freedom in their work, allowing them to focus their attention on new scientific breakthroughs.

logarithm Aid to calculation devised by John Napier in 1614, and developed by the English mathematician Henry Briggs. A number's logarithm is the power to which a base must be raised to equal the number, i.e. if b^x = n, then log_b n = x, where n is the number, b the base and x the logarithm. Common logarithms have base 10, and so-called natural logarithms have base e (2.71828…). Logarithms to the base 2 are used in computer science and information theory.