Example. and so on, until the successive numerators of the partial fractions corresponding to (a + bx)" are all of them determined. As an example, let it be required to resolve the In this case M=x2+2x+1, Q=x+4, and a=2: consequently, x2+2x+1 3 3 1 = + + x3-12x+16 2(x-2)2 4(x-2)4(x+4) ' The application of this method of resolving a fraction with a compound denominator into partial and more simple fractions, pre-supposes a knowledge of one or more of the component factors of its denominator; and the resolution will be complete when all those factors are giyen or discoverable: if, however, the denominator of the primitive fraction be given, without its factors, their discovery, as well as their necessary existence, is inseparably connected with the general theory of equations and of their solution. CHAP. XIV. Definition of the term logarithm. System of logarithms. Napierian logarithms. Tabular logarithms. ON LOGARITHMS AND LOGARITHMIC TABLES, and their 678. In the equation at =n, it is usual to give the general name of logarithm, Art. 364, to the index of that power of the same symbol a, which is equal to another symbol or quantity n: or in other words, a is called the logarithm of n to the base a. 679. A system of logarithms is the series of indices of the same base which correspond to the succession of values of n. 680. Thus, if the base be e=2.7182818..., then the corresponding indices form the Napierian system of logarithms, inasmuch as this was the base which was adopted by the inventor of logarithms, Lord Napier: it is this system which is almost exclusively used in algebraical formulæ; and the abbreviated form log n, always means the Napierian logarithm of n, unless the contrary be expressed, or be inferred from its position and usage. 681. If the base be the number 10, which is the radix or base of our scale of numeration and arithmetical notation, then the corresponding indices are called tabular logarithms, being such as are recorded in our ordinary tables of logarithms, and which are exclusively used in arithmetical calculations. 682. The term log n, when used in such calculations, Mode of designating would always mean the tabular logarithm of n, unless the tabular locontrary was expressed: for its position and usage would, garithms. under such circumstances, determine its meaning, without the aid of any specific designation: in other cases, it might be conveniently designated by t log, where the letter t is prefixed to log, to indicate that it is the tabular logarithm which is thus designated. of other systems. 683. The quantities e and 10 are the only bases of Logarithms logarithms which are used in analytical enquiries, though the general symbol a is employed to designate any base whatever, whether different from e and 10, or the same, the use of which may seem more adapted to the generality of algebraical investigations. arithmetical 684. If in the equation an, there are more sym- There is bolical values of a than one, which satisfy the required only one conditions, they are equally logarithms of n: it is very value of the logarithm easily shewn, however, that there is only one arithmetical of the same value of a which makes a* equal to an arithmetical value number or of n for if y be such a value, different from a, then quantity. a* a*=n and an, and, therefore, = @"=1, an equa a" tion which can only be arithmetically satisfied by making x-y=0; or, in other words, a is equal to y and identical with it we shall confine our attention, in the first instance, entirely to such arithmetical logarithms, and afterwards proceed to consider symbolical logarithms, as distinguished from the former. 685. The properties of logarithms are the properties of Properties of logathe indices of the same symbol: thus if an, and a"=n', rithms. then a+"=nn', a"-" == n 1 77,, a11=n", and ar=n: there n fore, if a be denoted by log' n (where log' is different from log or t log, unless ae or a=10), and a' by log'n, we shall have The term logarithm is used strictly in the sense of this definition, without any reference to the very limited meaning of it which is noticed in Art. 364. Examples. (1) Log'nn'=log' n + log' n', or the logarithm of the product of two numbers or quantities, is equal to the sum of the logarithms of the two factors, and conversely. (2) Log' n n = log'n-log'n', or the logarithm of the quotient of two numbers or quantities, is the logarithm of the dividend diminished by the logarithm of the divisor, and conversely. (3) Log'n' =p log' n, or the logarithm of the pth, or any power of a number is found by multiplying p, or the index of the power, into the logarithm of the number, and conversely. 1 1 (4) Log' n2 = - log'n or the logarithm of the pth, Р or any root of a number is found by dividing the logarithm of the number by the number which expresses the denomination of the root, and conversely. 686. Thus if n=37, and n' = 185, then we shall find from the tables, ()t log 3733 t log 37=4.7046051=t log 50653. |