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CITY OF

PLANE TRIGONOMETRY

CHAPTER I.

LOGARITHMS: REVIEW OF TREATMENT IN ARITH METIC AND ALGEBRA.

1. There is a large amount of computation necessary in the solution of some of the practical problems in trigonometry. The labour of making extensive and complicated calculations can be greatly lessened by the employment of a table of logarithms, an instrument which was invented for this very purpose by John Napier (1550–1617), Baron of Merchiston in Scotland, and described by him in 1614. From Henry Briggs (1556–1631), who was professor at Gresham College, London, and later at Oxford, this invention received modifications which made it more convenient for ordinary practical purposes.*

Every good treatise on algebra contains a chapter on logarithms. This brief introductory review is given merely for the purpose of bringing to mind the special properties of logarithms which make them readily adaptable to the saving of arithmetical work. A little preliminary practice in the use of logarithms will be of advantage to any one who intends to study trigonometry. A review of logarithms as treated in some standard algebra is strongly recommended.

* The logarithms in general use are known as Common logarithms or as Briggs's logarithms, in order to distinguish them from another system, which is also a modified form of Napier's system. The logarithms of this other modified system are frequently employed in mathematics, and are known as Natural logarithms, Hyperbolic logarithms, and also, but erroneously, as Napierian logarithms. See historical sketch in article Logarithms (Ency. Brit. 9th edition), by J. W. L. Glaisher.

(1)

a* =

N,

2. Definition of a logarithm.

If

then x is the index of the power to which a must be raised in order to equal N...

For some purposes, this idea is presented in these words: If athen x is the logarithm of N to the base a.

The latter statement is taken as the definition of a logarithm, and is expressed by mathematical symbols in this manner, viz.:

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Equations (1), (2), are equivalent; they are merely two different ways of stating a certain connection between the three quantities a, x; N. For example, the relations

$38, 5625, 10-100.001,

may also be expressed by the equivalent logarithmic equations, log, 83, log, 6254, log10.001 - 3.

EXAMPLES.

1. Express the following equations in a logarithmic form:

=

38 = 27, 44 = 256, 112 = 121, 93 728, 73 = 343, mb = p. 2. Express the following equations in the exponential form : log2 83, log5 625 = 4, log10 1000

=

3, log2 64

=

6, logn P= a.

3. When the base is 2, what are the logarithms of 1, 2, 4, 8, 16, 32, 64, 128, 256 ?

4. When the base is 5, what are the logarithms of 1, 5, 25, 125, 625, 3125 ? 5. When the base is 10, what are the logarithms of 1, 10, 100, 1000, 10,000, 100,000, 1,000,000, .1, .01, .001, .0001, .00001, .000001 ?

6. When the base is 4, and the logarithms are 0, 1, 2, 3, 4, 5, what are the numbers ?

7. When the base is 10, between what whole numbers do the logarithms of the following numbers lie: 8, 72, 235, 1140, 3470, .7, .04, .0035 ?

3. Properties of logarithms. Since a logarithm is the index of a power, it follows that the properties of logarithms must be derivable from the properties of indices; that is, from the laws

2.]

PROPERTIES OF LOGARITHMS.

3

of indices. The laws of indices are as follows (a, m, n, being any finite quantities):

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[If P=a", then log, P= p, MNP = am+n+

whence, log. MNP=m+n+p=log. M+ log, N+ log, P.]

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Also, M(a)=ar; whence, loga M" = rm = r loga M.、 (5)

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The results (3)-(6) state the properties, or are the laws of logarithms. They may be expressed in words as follows:

(1) The logarithm of the product of any number of factors is equal to the sum of the logarithms of the factors.

(2) The logarithm of the quotient of two numbers is equal to the logarithm of the numerator diminished by the logarithm of the denominator.

(3) The logarithm of the rth power of a number is equal to r times the logarithm of the number.

(4) The logarithm of the rth root of a number is equal to 1th of

the logarithm of the number.

Hence, if the logarithms (i.e. the exponents of powers) of numbers be used instead of the numbers themselves, then the operations of multiplication and thivision are replaced by those of addition and subtraction, and the aperations of raising to powers and extracting roots, by those of multiplication and division.

4. Common system of logarithms. Any positive number except 1ay be chosen as the base and to the base chosen there corresponds a set or system of logarithms. In the common or decimal system the base is 10 and, as will presently appear, this system is a very conventet one for ordinary numerical calculations. what follows, the base 10 is not expressed, but it is always understood that 10 is the base. The logarithm of a number in the common system is the answer to the question: "What power of 10 is the number ? "

Since

1=10°, 10 = 101, 100=102,

it follows that

1000=103,

10000=10', ...,

log 10, log 10= 1, log 100= 2, log 1000= 3, log 10000= 4,

This also shows that the logarithms of numbers

between 1 and 10 lie between 0 and 1,

between 10 and 100 lie between 1 and 2,

between 100 and 1000 lie between 2 and 3, and so on.

For example,

(1)

....

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Most logarithms are incommensurable numbers. (See Art. 9.) The decimal part of the logarithm is called the mantissa, the

The base of the natural system of logarithms is an incommensurable number, which is always denoted by the letter e and is approximately equal to 2.7182818284.

4, 5.]

COMMON LOGARITHMS.

5

integral part of the logarithm is called the index or charac teristic.

The two great advantages of the common system, as will now be shown, are:

(1) The characteristic of a logarithm can be written on mere inspection;

(2) The position of the decimal point in a number affects the characteristic alone, the mantissa being always the same for the same sequence of figures.

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.001 == 10-3, .000100010,; it follows that

log.11, log .012, log .0013, log .00014, etc. (2)

From (1) and (2) comes the following rule for finding the chur

acteristic:

When the number is greater than 1, the characteristic is positive and is one less than the number of digits to the left of the decimal point; when the number is less than 1,,the characteristis is negative, and is one more than the number of zeros between the decimal point and the first significant figure.

When a change is made in the position of the decimal point in a number, the value of the number is changed by some integral power of 10. Its logarithm is then changed by a whole number only, and, consequently, its mantissa is not affected. For example,

25.38 2538 x 10-2,

=

2538000 = 2538 × 103;

and hence, log 25.38 = log 2538-2, log 2538000= log 2538 +3.

Accordingly, it is necessary to put only the mantissas of sequences of integers in the tables.

5. Negative characteristics. In common logarithms the mantissa is always kept positive. Thus, for example, log 25380 = 4.40449; log .002538 log 2538 - log 1000000 = 3.40449 6 =-3+.40449. (Never put - 2.59551.)

=

2538 log

1000000

=

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