# 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

 I logb (xy) = logb x + logb y bn•bm = bn+m II log (x/y) = logb x - logb y bn/bm = bn+m III logb xy = y•logb x (bn)m = b(nm) IV logb x = (logb a)(logax) If x = br ; b = ap, then x = apr V logb bn = n If bn = bm, then n = m VI log 1 = 0 (any base) b0 = 1

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). Logarithmic scale. Illustration by Hans & Cassidy. Courtesy of Gale Group.

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.