LOGARITHMS AND EXPLANATION OF TABLES

1. Use of logarithms. By the use of logarithms, the processes of multiplication, division, raising to a power, and extracting a root, of arithmetical numbers are usually much simplified. The process of multiplication is replaced by one of addition, that of division, by one of subtraction, that of raising to a power by a simple multiplication, and that of extracting a root, by a division. Many calculations that are difficult or impossible by other mathematical methods are readily carried out by means of logarithms. It was said by the great French astronomer, Laplace, that the method of logarithms by reducing to a few days the labors of many months, doubled, as it were, the life of an astronomer, besides freeing him from the errors and disgust inseparable from long calculations. Of course these same advantages are shared by others who find it necessary to do numerical calculations.*

2. Exponents. The student is already familiar with the following definitions and theorems from algebra, concerning exponents. For convenience they are restated here.

* "The miraculous powers of modern calculations are due to three inventions: the Hindu Notation, Decimal Fractions, and Logarithms." Cajori, A History of Elementary Mathematics.

3. Definitions. If three numbers N, b, and x have such values that

N = b2,

then x is called the logarithm* of N to the base b. In words this gives the following.

DEFINITION. The logarithm of a number to a given base is the exponent by which the base must be affected to produce that number. If in the equation, N = b2, all possible positive values are given to N, while b is some positive number other than 1, the corresponding values of x form a system of logarithms.

4. Notation. If 4 is taken as a base, then, in the language, or notation, of exponents,

43
43 = 64.

In the language, or notation, of logarithms, the same idea is expressed by saying the logarithm of 64 to the base 4 is 3. This is abbreviated and written

* The word logarithm is derived from logos meaning ratio and arithmos meaning number.

number other than 1 may be used as a base for a system of logarithms, in practice only two bases are used.

(1) The common system, or Briggs' system, of which the base is 10.

(2) The natural system, also called the hyperbolic, or Napierian system, of which the base is a number that to seven decimal places is 2.7182818. This base is usually represented by the letter e.

The common system is the one commonly used in computing, and the natural system in more advanced and theoretical work. 6. Properties of logarithms. The use of logarithms depends upon the following properties, which are true for any base greater than unity:

(2) The logarithm of the base of any system is unity. Since b1b, for any base, log b = 1.

*Logarithms were invented by John Napier, Baron of Merchiston, of Scotland, who lived from 1550 to 1617. They were described by him in 1614. A contemporary of Napier's, Henry Briggs (1556 to 1631) professor of Gresham College, London, modified the new invention by using the base 10, and so made it more convenient for practical purposes. See Cajori, A History of Elementary Mathematics, page 160, et seq. For a very complete account of logarithms, see History of the Logarithmic and Exponential Concepts, by Florion Cojori, appearing in the American Mathematical Monthly, January to June, 1913.

(3) The logarithm of zero in any system whose base is greater than 1 is negative infinity.

(b)

흐흐

Nx M = b × by = bx+y;

x

.. log (N × M) = x + y = logɩ N + logɩ M.

N÷M = b2 ÷ by = bx-y;

bx

(c)

Nn = (bx)n

n log, N.

(d)

b n

; :: log VN

X

log, N.

The following theorems are therefore established:

(4) The logarithm of a product equals the sum of the logarithms of the factors. By (a).

(5) The logarithm of a quotient equals the logarithm of the dividend minus the logarithm of the divisor. By (b).

(6) The logarithm of a power of a number equals the logarithm of the number multiplied by the exponent of the power. By (c).

(7) The logarithm of a root of a number equals the logarithm of the number divided by the index of the root. By (d).

The truth of the statements in Art. 1 follows from these theorems. That is, the process of multiplication is replaced by an addition; division by a subtraction; raising to a power, by a multiplication; and extracting a root, by a division.

7. Logarithms to the base 10. In what follows, if no base is stated, it is understood that the base 10 is used.

When the base is 10 we evidently have the following:

It is evident that these are the only numbers between 0.00001 and 100,000 which have integers for logarithms. Every other number in this range has then for a logarithm an integer plus or minus a fraction. This fraction is put in the form of a decimal.

For instance, the logarithm of any number between 1000 and 10,000 is between 3 and 4, or it is 3+ a decimal. For a number between 100 and 1000 the logarithm is 2+ a decimal. Between 0.01 and 0.1, the logarithm may be 2+ a decimal or -1- a decimal; but, in order that the fractional part of the logarithm may always be positive, it is agreed to take the logarithm so that the integral part only is negative.

Usually, then, the logarithm of a number consists of two parts, an integer and a fraction, the fraction being the approximate value of an irrational number.

The integral part is called the characteristic.

The fractional part is called the mantissa.

The logarithm is the characteristic plus the mantissa.

The mantissas of the positive numbers arranged in order are called a table of logarithms.

The logarithm of 3467 consists of the characteristic 3 plus some mantissa because 3467 lies between 1000 and 10,000. The logarithm of 59,436 is 4 + a decimal because 59,436 lies between 10,000 and 100,000. The log 0.0236 = −2+ a decimal because 0.0236 lies between 0.01 and 0.1.

It is readily seen that multiplying a number by 10" increases the characteristic by n, where n is an integer; and dividing a number by 10" decreases the characteristic by n.

For, log (NX10") = log N+log 10"=log N+n log 10=log N+n, and log (N 10") = log N-log 10"-log N-n log 10=log N-n. This establishes the following:

THEOREM. The position of the decimal point in the number affects the characteristic of the logarithm only, the mantissa remaining unchanged for the same sequence of figures.