in this case, taken equal to 1, the above formula will then become From which it follows, that the logarithm of the nth root of any number is equal to the nth part of the logarithm of that number. Hence, besides the use of logarithms in abridging the operations of multiplication and division, they are equally applicable to the raising of powers and extracting of roots ; which are performed by simply multiplying the given logarithm by the index of the power, or dividing it by the number denoting the root. But, although the properties here mentioned are common to every system of logarithms, it was necessary, for practical purposes, to select some one of them from the rest, and to adapt the logarithms of all the natural numbers to that particular scale. And as 10 is the base of our present system of arithmetic, the same number has accordingly been chosen for the base of the logarithmic system, now generally used. So that, according to this scale, which is that of the common logarithmic tables, the numbers have for their logarithms . . . —4, -3, -2, -1, 0, 1, 2, 3, 4, &c. Which are evidently a set of numbers in arithmetical progression, answering to another set in geometrical progression; as is the case in every system of logarithms. And therefore, since the common, or tabular, logarithm of any number (n) is the index of that power of 10, which when involved, is equal to the given number, it is plain, from the following equation, : - 1 ... ." 10−n, or 10-5, that the logarithms of all the intermediate numbers, in the above series, may be assigned by approximation, and made to occupy their proper places in the general scale. It is also evident, that the logarithms of 1, 10, 100, 1000, &c. being 0, 1, 2, 3, &c. respectively, the logarithm of any number, falling between 0 and 1, will be 0 and some decimal parts ; that of a number between 10 and 100, 1 and some decimal parts ; of a number between 100 and 1000, 2 and some decimal parts ; and so on, for other numbers of this kind. - 1 100° so &c. or of their equals . 1, .01, .001, &c. in the descending part of the scale, being – 1, -2, -3, &c. the logarithm of any number, falling between 0 and 1, will be – 1, and some positive decimal parts; that of a number between .1 and .01, -2 and some positive decimal parts; of a number between .01 and .001,–3, and some positive decimal parts ; &c. - And for a similar reason, the logarithms of # Hence, likewise, as the multiplying or dividing of any number by 10, 100. 1000, &c. is performed by barely increasing or diminishing the integral part of its logarithm by 1, 2, 3, &c. it is obvious that all numbers, which consist of the same figures, whether they be integral, fractional, or mixed, will have, for the decimal part of their legarithms, the same positive quantity. * So that, in this system, the integral part of any logarithm, which is usually called its index, or characteristic, is always less by 1 than the number of integers which the natural number consists of ; and for decimals, it is the number which denotes the distance of the first significant figure from the place of units. Thus, according to the logarithmic tables in common Where the sign — is put over the index, instead of before it, when that part of the logarithm is negative, in order to distinguish it from the decimal part, which is always to be considered as +, or affirmative. Also, agreeably to what has been before observed, the logarithm of 38.540 being 4.58591.17, the logarithms of any other numbers, consisting of the same figures, will be as follows: Logarithms. 3.58591 17 2.58591 l'7 1.58591.17 0.58591.17 1.58591.17 2.5859 117 3.58591.17 Which logarithms, in this case, as well as in all others of a similar kind, whether the number contains ciphers or not, differ only in their indices, the decimal, or positive part, being the same in them all. (e) (e) The great advantages attending the common, or Briggean system of logarithms, above all * arise chiefly from the readi And as the indices, or integral parts, of the logarithms of any numbers whatever, in this system, can always be thus readily found from the simple censideration of the rule above mentioned, they are generally omitted in the tables, being left to be supplied by the operator, as occasion requires. It may here, also, be farther added, that, when the logarithm of a given number, in any particular system, is known, it will be easy to find the logarithm of the same number in any other system, by means of the following equations, ar:=n, and ex'=n ; or log, n=2, and l. n=a'. Where log denotes the logarithm of n, in the system of which u is the base, and l. its logarithm in the system of which e is the base. mess with which we can always find the characteristic or integral E. of any iogarithm from the bare inspection of the natural num er to which it belongs, and the circumstance, that multiplying or dividing any number by 10, 100, 1000, &c only influences the characteristic of its logarithm, without affecting the decimaal part. Thus, for instance, if i be made to denote the index, or integral part of the logarithm of any number N, and d its decimal part, we shall have log N= i+d; iog. 10m x N-(i+m)+ d ; log. H= (i-m) + d , where it is plain that the decimal part of the Frn logarithm, in each of these cases, remains the same. presses the constant relation which the logarithms of n have to each other in the systems to which they belong. But the only system of these numbers, deserving of notice, except that above described is the one that furnishes what have been usually called hyperbolic or Neperian logarithms, the base. e of which is 2.71828.1828 459. . . . Hence, in comparing these with the common or tabular logarithms, we shall have, by putting a in the latter of the above formulae = 10, the expression o or l. n=log. n Xl. 10. Where log. in this case, denotes the common tabular logarithm of the number n, and l. its hyperbolic loga - - - - l - - rithm ; the constant factor, or multiplier, II, which is 1 2,3025850929 being what is usually called the modulus of the common system of logarithms. (f) ProBLEM I. To compute the logarithm of any of the natural numbers 1, 2, 3, 4, 5, &c. § It may here be remarked, that, although the common logarithms have superseded the use of hyperbolic or Neperian logarithms, in all the ordinary operations to which these numbers are generally applied, yet the latter are not without some advantages peculiar to themselves; heing of frequent occurrence in the application of the Fluxionary Caicrlos, to many analytical and physical problems, where they are required for the finding of certain fluents, which could not be so readily determined without their assistance; on which account, great pains have been taken to calculate tables of hyperbolic logarithuns, to a co’s...", " .. cxo~nt, chiefly for this purpose Mr Barlow, in a too, on 2 thematical Tables lately published, has given them for the first 1000° numbers, - - |