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CHAPTER XIX

LOGARITHMS

150. The logarithm of any number is the exponent of the power to which a fixed number must be raised to equal the given number. The fixed number is called the base of the system of logarithms.

The Briggs or Common system of logarithms uses the number 10 as its base. Hence, the common logarithm of any number is the exponent of the power to which 10 is raised to equal the given number. Thus since 100 = 102, the logarithm of 100 is 2, which is written, log 100 = 2. In this book, the common system of logarithms only is used. Since 100 = 1 . log 1 = 0

10' = 10 i log 10 = 1
102 = 100 i log 100 = 2
103 = 1000 log 1000 = 3
104 = 10000 i. log 10000 = 4

and so on.

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Hence, the logarithms of numbers greater than 1 are positive, and the logarithms of numbers less than 1 are negative. Also, since most numbers are not an integral power of 10, the logarithms of such numbers are not integers, but mixed numbers, consisting of an integral part and a fraction. This fractional part of the logarithm is usually expressed as a decimal.

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JOHN NAPIER (From an Engraving After an Original Edinburgh Painting) Born near Edinburgh, 1550. Died, 1617.

Inventor of logarithms.

His Mirifici logarithmorum canonis descriptio (1614), the first important work on mathematics produced in Great Britain, inspired Briggs to develop the common system of logarithms with the decimal base.

Napier was a student of theology and astrology, engaging in many controversies.

The great object of his mathematical studies was to simplify and systematize arithmetic, algebra and trigonometry.

Napier succeeded in constructing a system of logarithms before exponents were used.

The integral part of the logarithm is called the characteristic, and the decimal part is called the mantissa of the logarithm.

Since the log 1 = 0 and log 10 = 1, the logarithms of the numbers between 1 and 10 must lie between 0 and 1, that is, the logarithm of any number, having one figure in the whole number, is some decimal fraction.

Similarly, since log 10 = 1 and log 100 = 2, the logarithms of numbers between 10 and 100 must lie between 1 and 2, that is, the logarithm of any number, having two figures in the whole number, is 1 plus some decimal. In the same way it may be shown, that the logarithm of any number having three figures in the whole number is 2 plus some decimal.

Hence, the following rule:

The characteristic of the logarithm of any number greater than 1 is positive and 1 less than the number of figures in the integral part of the number.

Note that it is the integral part of the logarithm which is called the characteristic. An ordinary number has no characteristic. Thus, if the log 32.56 = 1.51268, 1 is the characteristic of the logarithm, while 32 is simply the integral part of the number 32.56.

Since the log 1 = 0 and log .1 = -1, the logarithms of numbers between 1 and .1; lie between 0 and -1, that is, the logarithm of any decimal fraction, whose first significant figure is in the first place of decimals, is a negative decimal.

Similarly, since the log .1 = -1 and log .01 = -2, the logarithms of numbers between .1 and .01 lie between

- 1 and — 2, that is, the logarithm of any decimal fraction whose first significant figure is in the second place of decimals is – 1 plus some negative decimal.

Instead of writing these logarithms entirely as negative numbers, it is found more convenient to express these logarithms in the form of a negative characteristic plus a positive decimal fraction; that is, the mantissa is always positive.

Thus, the log .03256 = -1.48732 but is expressed as -2 + .51268 which is equal to – 1.48732.

Therefore, the logarithm of any number between 1 and .1 has the form –1 + some positive decimal, the logarithm of any number between .1 and .01 has the form –2 + some positive decimal and so on. Hence, the following rule:

The characteristic of the logarithm of any number less than 1 is negative and equal to the order of the place of the first significant figure.

Thus, the characteristic of the logarithm of .00456 is – 3, since the first significant figure, 4, is in the third place of decimals. · For convenience, a negative characteristic is usually written in two parts by adding and subtracting 10 or some multiple of 10.

Thus, the logarithm -1 +.52368 is written 9.52368 – 10, 9 – 10 being equal to -1. .

Since all logarithms are exponents, they can be used only in arithmetical operations that can be performed by means of exponents, as explained in the chapter on Theory of Exponents. Logarithms may therefore be used in any computation involving multiplication, division, involution or evolution.

If the number 32.56 = 101.51268, log 32.56 = 1.51268. Then 32.56 X 10 = 10.1.51268 X 101 = 102.51268 that is, 325.6 = 102.51268 .. log 325.6 = 2.51268. Also, 32.56 · 1000 = 101.51268 • 103 = 10–2+.51 268, that is, .03256 = 10–2+.51268 .. log .03256 = –2 + .51268.

This shows that a multiplication or division by any power of 10 does not change the mantissa of the logarithm. But a multiplication or division of any number by a power of 10 merely shifts the position of the decimal point, to the right in multiplication, to the left in division.

Hence, the mantissa is not changed by the shifting of the decimal point, that is, the value of the mantissa depends only on the combination of figures in the number, and will

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remain constant, whether the combination of figures represents a whole number, a mixed number, or a decimal fraction.

Since the characteristic of the logarithm of any number can be determined by inspection according to the rules given, it is only necessary to find the mantissa.

The mantissas for various combinations of figures have been computed and may be found in the table of logarithms, page 361.

USE OF THE TABLE OF LOGARITHMS 151. This table of logarithms gives the mantissa, to five decimals, for any combination of four figures.

The first three figures are given in the column at the left of the page, headed No, and the fourth figure is placed at the top of the ten columns, 0 to 9.

1. Find log 3743. The characteristic of the log 3743 is 3, one less than the number of figures in the whole number.

On page 366 of the table, we find the first three figures 374 in column No, and in line with 374, in column headed 3, we find the mantissa 57322, hence log 3743 = 3.57322.

2. Find log 57.64.
The characteristic of log 57.64 is 1.

The mantissa is found on page 370, in line with 576, column headed
4, 76072.
Hence, log 57.64 = 1.76072.
3. Find log. .001236.
The characteristic of log .001236 is – 3, written 7 – 10.

The mantissa for the combination 1236 is found on page 361, in line 123, column headed 6, 09202.

Hence log .001236 = 7.09202 – 10.
4. Find log 4.7757.
The characteristic of log 4.7757 is 0.

As our combination consists of five significant figures, and only four are given in the table, a process, called interpolation, is necessary.

On page 368, we find the mantissa for 4775, the first four figures of the combination, to be .67897.

The mantissa for 4776 is .67906. Hence, for an additional unit in the combination of four figures, the mantissa is increased by

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