214. A system is rendered determinate by fixing, arbitrarily, the number to which a particular logarithm is to correspond; as, for example, by assigning the number whose logarithm shall be 1. In every system the number whose logarithm is 1 is called the base of that system. In Napier's system, the base or number whose logarithm is 1 is 2-718281828. The base of the system of the common tabular logarithms is 10. Let Napier's logarithm of the number a be denoted by La; and the logarithm of a in any other system by log. a; then Mx La= log. a, and, therefore, M= log.a Whence the modulus of any system of logarithms is obtained by dividing the logarithm of any number a in that system by Napier's logarithm of the number a. Therefore the logarithms of Napier being given, if the logarithm of one number in any other system is known, the modulus, and by consequence the logarithms also, of that system can be found. Since the logarithm of the base of every system is 1, if the base of any system is denoted by a, the formula, M= log. a La becomes M=1a Consequently the modulus of any system is equal to the reciprocal of Napier's logarithm of the base of that system. Napier's log. of 10 being 1 2.302585092, or 434294481 is the modulus of the common 2.302585092 tabular logarithms. The tabular logarithm of any number can consequently be obtained by multiplying Napier's log. of that number by 434294481. 215. In the system whose base is 10, the logarithms of 10, 100, 1000, &c. are 1, 2, 3, &c., and, in general, the log. of 10=m. Of numbers falling between 1 and 10, the logs. fall between 0 and 1, 10 and 100 100 and 1000 1 and 2, 2 and 3, The integer part of a logarithm is named the characteristic, and the decimal part the mantissa. It may, therefore, be concluded that in this system the characteristic of the log. of a number contains as many repetitions of 1, less one, as the number contains figures. ... If a number is multiplied by 10, 100, 1000, . . . its log. is augmented by 1, 2, 3, . . . and if a number is divided by 10, 100, 1000, its logarithm is diminished by 1, 2, 3 ... Therefore, if a number is multiplied or divided by a power of 10, so long as the result is greater than 1 the mantissa is constant, and the characteristic only varies. Taking the number 658, the log of which is 2.818226, Log. (65810 or 65.8)= log. 658-log. 10-2.818226-1=1.818226. From these instances it appears that when the divisor is a power of 10, and the result of the division is less than 1, the log. becomes negative, and all the figures are changed. In this case, instead of subtracting the whole log. of the dividend from the log. of the divisor and affecting the remainder with the sign it is usual to subtract only the characteristic of the log. of the dividend from the log. of the divisor, and to consider the remainder as a negative characteristic. The negative sign is not prefixed to, but placed over, the characteristic, thus, 2-818226-3 is written I-818226; 2.818226-4 expressions which are to be considered as abbreviations, the first of 0.818226-1 or of -1+0.818226, and the second of 0.818226-2 or of -2+0.818226. By admitting, in this manner, logarithms whose characteristics alone are negative, it becomes permissible to say that, a decimal number being given, the place of the decimal point may be changed at pleasure without change of the decimal part of the logarithm. 216. In the system whose base is 10, the logarithms of the series of whole numbers, from 1 to 10000, for example, may be computed in the following manner. Since 1 and 10 are two terms of the geometrical progression, and 0, 1 the corresponding terms of the arithmetical progression, if a great number of geometrical means is inserted between 1 and 10, and the same number of arithmetical means between 0 and 1, and the two progressions are extended until the first reaches 10000; then certain terms of the first progression differ from the numbers 2, 3, 4, . . . . by less than any assigned number (Art. 212). The corresponding terms of the second progression differ from the logarithms of these numbers by less than any assigned number. For, let N be a whole number which falls between two terms n, n' of the geometrical progression, and let the corresponding terms of the arithmetical progression or the logarithms of n, n' be l, l'; then if the number of geometrical and arithmetical means is more and more augmented, the terms of the two progressions are made nearly to increase continuously; and the number N falling between n, n' is found to have a corresponding term L between l and l. Now, since L falls between 7 and l', L—l<l'—l and l'-L<l'—l; therefore the error committed by taking either lor l' for the logarithm of N is less than the difference l'-l, which can be rendered less than any assigned number. 1 1000000' 1 1000000 If the ratio of the arithmetical progression is =000001; and the progression gives logarithms with six correct decimal figures. That a geometrical progression may correspond with an arithmetical possessing this degree of accuracy, 10 must have 1000000 terms before it. To obtain the ratio of this progression it is necessary to extract the 1000000th root of 10. Since 1000000=106=26 × 56, the extraction of the millionth root may be affected by repeated extractions of the 2d and 5th roots. 217. There is, however, a method of calculating logarithms without extracting a higher root than the second. Let n be the number whose logarithm is to be computed, n being between 1 and 10. 2 2' The geometrical mean between 1 and 10=√1× 10, let this be denoted by A; the arithmetical mean between 0 and 1= 0+1=1, let this be denoted by a. Suppose that n falls between 1 and A, let the geometrical mean between 1 and A=B, and the arithmetical mean between 0 and a=b. Suppose next that n falls between A and B, find C, the geometrical mean between A, B, and c, the arithmetical mean between a, b. This series of operations being continued until n is contained between two geometrical means, to which correspond two arithmetical means, of which the first six decimals are the same, it is certain that one of these means is the logarithm of n with the degree of approximation required. To attain the maximum of approximation with six decimals it is necessary to calculate the seventh decimal figure, and in the case in which the seventh figure is 5 or greater than 5 to add 1 to the sixth decimal figure of the logarithm; by this means the error in excess or defect is rendered less than the half of one millionth part of 1. The logarithms of prime numbers need alone to be computed, for the logarithms of other numbers are obtained by adding together the logarithms of the prime factors of which these numbers are composed. The error in the logarithm of a composite number may by these additions be increased to more than the half of one millionth. It is, however, easy to fix the maximum of error which can be produced by this means. In the present hypothesis the highest number in the tables is 10000; now the least prime number is 2, and 217=131072; therefore a number which is less than 10000, being less than 217, cannot have more than seventeen factors, consequently the error of each logarithm being less than the half of one millionth of 1, the sum of the errors, even when they are either all in excess or all in defect, must be less than seventeen half millionths. By calculating the logarithms of the prime numbers to eight decimal figures the maximum of error in the logarithm of the greatest composite number in the tables cannot exceed two units of the seventh order of decimals, and consequently it must be less than one unit of the sixth order. 218. When systems of logarithms are determined by their bases the transition from one system to another is easy; for, denoting by a a very small quantity, the series of numbers is given by the geometrical progression 1:1+a: (1+a)2 : (1+a)3, &c.; and supposing 3 and y to be also very small quantities, the logarithms of the two systems which are compared may be taken respectively in the two arithmetical progressions: +0.8.23.33.43. +0. y. 2y. 3y. 4y It is evident that the logarithms of the same number in the second system and in the first have to each other a constant ratio, so that denoting by r', r the two logarithms, and by K the constant ratio, x'=Kx. With respect to the ratio K, to obtain it it is sufficient that the logarithms of a single number are known in the two systems. Now in the first system the logarithm of the base in the second is supposed to be known, and in the second system the logarithm of its own base is 1; making, therefore, a' the second base, and denoting by logarithm a' the logarithm of the second base in the first system, 1 1 : : 1 · (1+a)3 · (1+a)2 · 1+a : 1 : (1+a) : (1+a)2 : (1+a)3 : . . . . -33 -23 -3.0. B. 2,3 3,3 if a, Bare indefinitely small quantities the terms of the progressions may be considered to vary continuously, and consequently the second series must contain a multiple of 6, which is equal to 1. Let uß be this multiple, and let the corresponding term of the first series be equal to a, then simultaneously pẞ=1, (1+a)"=a. from the comparison of which it is evident that if any term of the second progression is denoted by x and the corresponding term of the first progression by y, the relation y=a* must subsist between these terms. The logarithms of numbers may hence be defined as the exponents of the powers to which it is necessary to raise a constant quantity (called Base) in order to deduce from it all these numbers. In adopting this definition it is to be understood that the base is a real and positive quantity. Under this limitation the base may be any number different from 1. To establish this proposition, it is necessary to prove that if in the exponential equation, y=a*, all possible values, negative as well as positive, are given to the exponenta, the corresponding values of y will comprehend all the numbers continuously between zero and infinity. 1st. Let a be greater than 1. If to x are given positive and increasing 1 2 values (beginning from zero) such as 10' 10' &c., then, y=ao, a, a, a,... or making Va=a', y=1, a', a", a'3, a' being a quantity greater than 1, this series increases to infinity; and it is further evident that if x is made to increase by very small increments the consecutive values of y may be made to differ from each other by less than any given number. 1 It has been proved that if z is made to pass through all positive values, beginning from 0, a increases continuously from 1 to ∞, therefore the quotient must decrease from 1 to 0; whence it is inferred that the negative ∞ make y to take all descending values from a' values of x between 0 and Since a<1, a'>1, therefore according as x is made to increase positively or negatively a will increase from 1 to cc or decrease from 1 to 0, and consequently y will decrease from 1 to 0 or increase from 1 to ∞. It is to be concluded, therefore, that every number except 1 may be taken for the base of a system of logarithms. The preceding discussion also shows, 1st. That in every system of logarithms the logarithm of 1 is 0 and the logarithm of the base is 1. 2d. That in systems whose base is greater than 1, log. ∞ ∞ and log. 0= 3d. That in systems whose base is less than 1, log. ∞ ∞ and log. 0= x. 220. By the second definition of logarithms (Art. 219), the properties of logarithms are derived immediately from those of exponents. For if y, y', y',.. are numbers, and x, xx".. the logarithms of these numbers, by the definition y=a", y'=a*', y''=a*" ... And by the rules for the calculus of exponents, .... ... ... y"=(a*)"=""; √y=Va=a^. But by the definition (Art. 219), the exponents of a in the expressions a2 + x1 + x!!... a2-*', a", a", are the logarithms of y.y'.y' y .; y", "√yrey" spectively; whence, as has been already proved (Art. 208-211), the logarithm of a product is equal to the sum of the logarithms of the factors of that product; the logarithm of a quotient is equal to the excess of the logarithm of the dividend over the logarithm of the divisor; the logarithm of a power is equal to the product of the logarithm of the root by the exponent of the power; and the logarithm of a root is equal to the quotient of the logarithm of the power by the index of the root. 221. Let it be required to calculate the logarithm of a given number b in the system whose base is a given number a. This problem involves the resolution of the equation a=b, in which the unknown quantity is an exponent. To resolve it. Suppose that the numbers a, b are both greater than 1; by giving to x the values 0, 1, 2, 3, &c. two powers of a, which may be represented by a" and a"+1, can be found between which b is comprehended; then x is between n and n+1. ... In this expression c>l and c<a, for b>a" and b<a"+1, consequently in forming the powers c1, c2, c3. it is found that a is comprehended between two consecutive powers c" and "+1, and that is therefore between n' and n'+1, Operating in the same manner on this equation, it is found between what whole numbers n" and n"+1, x" is comprehended; then " being made equal to n"+ the process may be repeated as often as necessary. |