CHAPTER XV

LOGARITHMS

SECTION 1, LOGARITHM OF A NUMBER GREATER THAN UNITY. SECTION 2, LOGARITHM OF A NUMBER LESS THAN UNITY. SECTION 3, NAPERIAN OR HYPERBOLIC LOGARITHMS. SECTION 4, LOGARITHM OF A PRODUCT. SECTION 5, LOGARITHM OF A QUOTIENT. SECTION 6, LOGARITHM OF A POWER. SECTION 7, LOGARITHM OF A ROOT. SECTION 8, SOLUTION OF AN EXPONENTIAL EQUATION. SECTION 9, MODEL SOLUTIONS. SECTION 10, LOGARITHMIC COMPUTATION.

228. Two Ways of Multiplying. 100 may be multiplied by 1000 as follows:

(1)
(2)

100X1000=100000,

102 X 103-105 = 100000.

Observe that in (2) the product is obtained by addition of the exponents of the powers of 10 which equal 100 and 1000. It is therefore possible to multiply together numbers which are integral powers of 10, by addition of the exponents of these powers.

In like manner 472 may be multiplied by 67.5 by addition of the exponents of the powers of 10 which equal 472 and 67.5. But here are two difficulties:

What powers of 10 equal 472 and 67.5, and what does 10 equal when raised to the sum of these two powers? The answer to this question and the removal of the difficulties follow:

100=102 472=102+

1000=103

67.5=101+

10=101

472 is greater than the 2d power of 10 and less than the 3rd; 67.5 is greater than the 1st power of 10 and less than the 2d.

A table of logarithms is an arrangement of numbers in sequence with the decimal parts of the powers of 10 which equal the numbers.

Taking the decimals from the table we have:

In practice the labor of this method of multiplication is reduced by setting down only the exponents of 10 as follows:

In like manner by the use of a table of logarithms, one number may be divided by another and any power or root of a number determined.

229. What a Logarithm is. Logarithms are used as a means of shortening and simplifying the mathematical processes of multiplication, division, powers, and roots.

In every system of logarithms all numbers are regarded as powers of another number which is called the base of the system.

Therefore the definition of a logarithm:

The logarithm of a number is the exponent of the power to which the base of the system must be raised to equal the number.

In the system of logarithms in common use, called the common or Briggs' System, the base of the system is 10. Hence in this system all numbers are regarded as powers of 10.

Consider any number, as 306. In the Briggs' system the logarithm of 306 is the exponent of the power to which the base 10 must be raised to equal 306.

=

Now 102 100 and 103=1000. But 306 is greater than 100 and less than 1000. Therefore in order to obtain 306 from 10, 10 must be raised to a power between the second and the third.

As shown 2+a decimal is the exponent of the power to which the base 10 must be raised to equal 306.

But by definition the exponent of the power to which the base must be raised to equal a given number is the logarithm of that number.

Therefore log 306=2+a decimal.

This decimal is given in the table of the Logarithms of Numbers.

§ 1. THE LOGARITHM OF A NUMBER GREATER THAN UNITY

230. A Number Having Three Figures. Direction I. In the table of the Logarithms of Numbers find 306 in the column headed N.

Place the index finger of the left hand directly under 306 and move the hand to the right in a horizontal line until it is under the number in the column headed 0. This number with the two figures prefixed (called leading figures) which are immediately above the blank space to the left of it in the same column, is the decimal part of the logarithm 306.

Therefore

102.485721 = 306;

=306; log 306=2.485721.

The log of 306 therefore consists of two parts:
(1) Integral, called the characteristic,

(2) Decimal, called the mantissa.

The decimal part only, is given in the tables.

231. Accurate Use of the Tables. The above direction for using the index finger of the left hand is given in order to secure speed and accuracy in the use of the tables. It makes possible the unobstructed use of the right hand for writing the figures from the tables and the index finger can be kept in its position on the page until the required number has been written from the table and the written number compared with the printed number.

Another excellent method is to move a straight-edge or a blank sheet of paper up or down the page until the required number can be read just above the upper edge.

232. Reason for the Characteristic. The integral part of the logarithm of 306 is 2, but the number of figures in 306 is 3.

In this instance the integral part of the logarithm is one less than the number of integral figures in 306, the natural number.

This is also true regarding any other number, for example 4798.

This number is greater than 103 and less than 101. Therefore as in the case of 306 the integral part of the logarithm is one less than the number of integral figures in the natural number.

233. Characteristic of the Logarithm of a Number Greater than Unity. Rule 1. The characteristic (integral part) of the logarithm of any number greater than 1 is one less in unit value than the number of integral figures in that number.

By this rule the characteristic of the logarithm of any number greater than unity may be determined without

the necessity of locating it with respect to integral powers of 10.

234. Position of Decimal Point. Since the division of a number by 10 is made by moving the decimal point one place to the left and division by 100 is made by moving the decimal point two places to the left and so on, the position of the decimal point affects the characteristic only.

Each number, being one-tenth the preceding, represents one less integral power of 10. Therefore, every shift of decimal point to the left means one less integral power of 10 and every shift to the right, one more integral power of 10.

In both cases the mantissa is the same. In reading mantissas, therefore, disregard decimal points in the natural numbers.

235. A Negative Characteristic. The mantissas of the logarithms of all numbers are positive but the characteristics may be positive or negative. They are positive for numbers greater than unity; they are negative for decimals.

It is therefore impossible to denote the logarithm of a decimal by a minus sign written in the usual position since that would indicate the entire logarithm as negative. Accordingly negative characteristics are indicated by a minus sign ABOVE them.

* The minus sign above the characteristic is used to denote that the characteristic is negative but that the mantissa is positive.