PLANE TRIGONOMETRY. CHAPTER I. LOGARITHMS: REVIEW OF TREATMENT IN ARITHMETIC 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. 2. Definition of a logarithm. If a* = N, (1) 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 a = N, then 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.: 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 10-3: 238, 5625, 10-3=1000= .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: 2. Express the following equations in the exponential form : 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 of indices. The laws of indices are as follows (a, m, n, being any finite quantities): MN= a+"; whence, loga MN=m+n=loga M+loga N. (3) [If P= a2, then log, P=p, MNP= am+n+p; whence, log. MNP=m+n+p=log. M+log. N+ loga P.] Also, M=(a)" amr; whence, loga M" = rm = r loga M. (5) Also, whence, = 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 division are replaced by those of addition and subtraction, and the operations of raising to powers and extracting roots, by those of multiplication and division. 4. Common system of logarithms. Any positive number except 1 may 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 convenient one for ordinary numerical calculations. In 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: power of 10 is the number ?" "What log 1= 0, 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, 9=10.95424 247=102.39270 1453 = 103.16227; or log 9.95424, log 247 2.39270, log 1453 = 3.16227. = (1) .... 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. |