THE PROPERTIES AND CONSTRUCTION OF LOGARITHMS. ART. 1. DEF. 1. In the equation u = a*, where a is a constant number greater than unity, and u any natural number, the index x is defined to be the logarithm of the number u to the base a. The notation assumed to denote "the logarithm of the number u to the base a' " is logu, so that x=logu, and the equation_u = a* may be written u= = alogau DEF. 2. The base of any system of logarithms is any fixed number which, being raised to the powers denoted by the logarithms, produces the successive natural numbers. DEF. 3. A system of logarithms is a series of the successive values of x derived from the equation u=a*, when the natural numbers 0, 1, 2, 3, 4, &c., are successively substituted for u, the same base a being preserved.1 1 Logarithms may be defined to be, as in fact they are, a series of numbers in Arithmetical progression which increase by a common difference, corresponding to another series in Geometrical progression which increase by a common multiplier. For example, let 10 be made the base, Then the numbers 0, 1, 2, 3, 4, &c., are the logarithms of the series of numbers 1, 10, 100, 1000, 10000 &c., respectively, to the base 10. Hence it is obvious that a negative number cannot be assumed as the base of any system of logarithms; for the odd powers of a negative number are negative, and the even powers are positive, and consequently they are not subject to the law of continuity in producing in order all the natural numbers. This definition of a system of logarithms suggests a method of finding the logarithms of all the intermediate numbers; for the Arithmetic mean between any two consecutive terms of the Arithmetic series will be the logarithm of the Geometric mean of the two corresponding terms of the Geometric series : Thus, the A. mean between 0 and 1 is 5, and the G. mean between 1 and 10 is 3.1622777; and the G. mean between 10 and 100 is 31 6227766; and so on for successive mean proportionals. Next, the A. means can be found between every two consecutive terms of the A. series 0, 5, 1, 1.5, &c., and the G. means between every two corresponding terms of the G. series 1, 3.1622777, 10, 31-6227766, 100, &c.; and so continuing the same process, may be found the logarithms of all numbers, but at very great expense of time and labour. A system of logarithms may be calculated to any base except unity, and hence there may be an indefinite number of systems of logarithms, according to the different assumptions made for the bases. There are, however, only two systems of logarithms used by mathematicians, one for shortening numerical calculations and the other in analytical reasonings. The following consequences may be shown to arise from the equation u alogal : The logarithm of 1 is 0, or log,1 = 0; and the logarithm of the base is 1, or log1a = 1. If logu be positive, and assume successively and continuously all possible values from 0 to +∞o it is obvious that u will receive all values from 1 to ∞. If log," be negative, and assume all possible values from 0 to, u will receive all values from 1 to 0. Hence, as logu changes continuously from to∞, u changes continuously from + to 0, and consequently produces all the positive natural numbers. If the base a be 10 and remain constant, and u be made to assume successively 1, 2, 3, 4, &c., the corresponding values of x in the equation u= 10, when computed and registered will form a table of that system of logarithms whose base is 10. 2. PROP. To find the logarithm of the product of two numbers. Here u,aloga, and u2aloga, by def. •'. U¡ . U2 = alogau1, alogau2 = aloɣau,+10gauz And log,{u. u2} =log11 + log12 by def. Or, the logarithm of the product of two numbers, is equal to the sum of the logarithms of the numbers themselves. COR. In a similar way it may be shewn that the loga{u. Ug. U3....} = log. + log12 + logз + .... Or that the logarithm of the product of any number of factors, is equal to the sum of the logarithms of the several factors. 3. PROP. To find the logarithm of the quotient of two numbers. Or, the logarithm of a quotient, is equal to the difference arising from subtracting the logarithm of the divisor from the logarithm of the dividend. ૫, Con. log. {-} = log."-log.", = — {log."— log.4} = — log. { "; } Or, the logarithm of any fraction is equal to the logarithm of its reciprocal taken negatively.' 4. PROP. To find the logarithm of any power of a number. Here u= alogau by def. Then raising each to the nth power. an = anlogau ... log.{u"} = n logu. Or, the logarithm of any power of a number, is equal to the product of the logarithm of the number and the index of the power. 5. PROP. To find the logarithm of any root of a number. Or, the logarithm of any root of a number, is equal to the quotient arising from dividing the logarithm of the number by the index of the root. Hence it appears that if a table of the logarithms of the natural numbers be arranged in order; by means of them can be performed. the operations of multiplication, division, involution and evolution of all numbers within the limits of the table. Thus, if one number is to be multiplied or divided by another, by taking their logarithms from the table, and adding or subtracting them, and then by finding in the table the number whose logarithm is equal to the sum or difference, the product or quotient of the two numbers is found. And the power or root of a number is found by taking the logarithm of the number from the table, and multiplying or dividing it by the index of the power or root, and then by finding in the table the number whose logarithm is equal to this product or quotient, the power or root of the proposed number is determined. Thus, by the aid of a table of logarithms, the arithmetical operations of multiplication and division may be effected by addition and subtraction: and those of involution or evolution by multiplying or dividing the logarithm by the index of the power or root. These are the advantages of logarithms in effecting numerical computations. и 1 Care must be taken not to confound the expressions log. {} and loga u former being the logarithm of the quotient of two numbers, which has been shewn equal to the difference arising from subtracting the logarithm of the divisor uz from the logarithm of the dividend u,; while the latter is the quotient arising from dividing the logarithm of u, by the logarithm of 2. DEF. The integral part of a logarithm is named its characteristic, the decimal part its mantissa.' In all arithmetical computations by logarithms, the mantissa is always positive, but the characteristic may be positive or negative. 6. Pror. To explain the advantages of that system of logarithms whose base is 10, the same as the radix of the scale of notation. By considering log...(10′′.u) and log..{1} ; log10".} = log1, 10"+log1 = n log1010 + log10u=n + log102, 10 and loge{}= log-log110* น That is, the logo{10".u} and logo {16- } are found from log10 ❝, 10" by simply increasing or diminishing the characteristic of log1ou by n. Hence, the logarithms of all numbers consisting of the same significant figures, whether integral, decimal, or partly integral and partly decimal, have the same mantissa; the only difference being in the value of the characteristic. 7. PROP. To find the law of the characteristics of that system of logarithms whose base is 10.2 Let any integral number u consist of n digits. It lies between 10′′-1 and 10"; 4 The word mantissa appears to be a Tuscan word, formerly employed in commerce, and meaning ever-measure or over-weight, "additamentum quod ponderi adjicitur.” The following logarithms of the prime numbers less than 100 are here given to enable the student to obtain numerical results in the exercises. In the printed tables of logarithms, the characteristics are omitted, and only the decimal parts are given without the decimal point. Of the Mathematical tables published by Dr. Hutton, one table calculated to seven places of decimals contains the logarithms of the natural numbers from 1 to 100,000. In the table published by Mr. Babbage, the Logarithms of the numbers are extended from 1 to 108,000, and very great care was taken by Mr. Babbage to secure the accuracy of them. Nos. Logarithms. Nos. Logarithms. Nos. Logarithms. If the number assumed for the base of a system of logarithms be the same as the radix of the system of notation employed, a great advantage arises; as in the system of notation whose radix is 10, the mantissa of any number composed of the same digits will have the same mantissa, whether the number be integral, or decimal, or and therefore the logarithm of u lies between n—1 and r, and consequently consists of n-1 units increased by some decimal: that is, the characteristic of log1ou is n-1. 10 Next, let u' be a decimal having n-1 ciphers between the decimal point and the first significant figure. .. the logarithm of u' lies between (n-1) and -n, and consequently consists of -n, increased by some positive decimal part: that is, the characteristic of logo' is n. Hence the general rule. For numbers wholly or partly integral, the characteristic is always less by unity than the number of integral places of which the number consists: and for decimals, the characteristic is the number (taken negative) which expresses the distance of the first significant figure of the decimal from the place of units. 8. PROP. The logarithm of a number less than 1, being negative, can always be expressed so that its mantissa shall be positive, and only its characteristic negative.1 Let u be a number less than 1, n the characteristic, m the mantissa of its logarithm. = Then logo= == (n + m) (n + 1) + (1 − m) of which 1 m is positive. Hence a logarithm wholly negative may be transformed into one whose characteristic only is negative, by increasing the negative characteristic by 1 and replacing the mantissa by its arithmetic complement, or its defect from 1. And conversely. A logarithm whose characteristic only is negative, may be transformed into a logarithm wholly negative by diminishing the characteristic by 1, and replacing the mantissa or decimal part by its arithmetical complement. partly integral and partly decimal, as will be seen in the logarithms of the numbers composed of the significant digits 6375. = 1.6989700, a logarithm with its mantissa positive and its characteristic negative. Ex. 2. Logiologio 5-logio 9, which is wholly negative. = = = (1 + log10 5)-1-logio 9=(log10 10+ log10 5)-1-log 9 |