consequently, x+x'+x", &c.=log. yy'y", &c.; or log. yy'y", &c.=log. y+log. y'+log. y", &c. The logarithm of the product of any number of factors is, therefore, equal to the sum of the logarithms of those factors. 506. Hence, if all the factors y, y', y", &c. are equal to each other, and the number of them be denoted by m, the preceding property will then become log. (ym)=m, log. y. Therefore the logarithm of the mth power of any number is equal to m times the logarithm of that. number. 507. In like manner, if the equation a*=y, be divided by a=y', we shall have, from the nature of X--X/ y or a -= ; and by the definition of y y ; or log. y-log. y'= y' logarithms, x-x'=log. log. Hence the logarithm of a fraction, or of the quotient arising from dividing one number by another, is equal to the logarithm of the numerator minus the logarithm of the denominator. 508. And if each member of the equation, a*=y, be raised to the fractional power, m we shall have m m ay"; and consequently, as before, x=log. (y") m m =log. Wym; or log. y= log. y. y' n n Therefore the logarithm of a mixed root, or power, of any number, is found by multiplying the logarithm of the given number, by the numerator of the index of that power, and dividing the result by the denominator. 509. And if the numerator m of the fractional index of the number y, be, in this case, taken equal to 1, the preceding formula will then become log. y=log. y. 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. 510. 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. 511. But, although the properties here mentioned are common to every system of logarithms, it was necessary for practical purposes to select some one of these systems from the rest, and to adapt the loga. rithms 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. 512. So that, according to this scale, which is that of the common logarithmic tables, the numbers, --4 --3 -2 etc. 10 10 10 10 ̈', 10°, 10', 103, 10 ̊, 10, etc. etc.; or, 1 1 1, 10, 100, 1000, 10000, 1 etc., have for their logarithms, etc. -4, -3, -2, -1, 0, 1, 2, 3, 4, etc. 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. 513. 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 equation 10"=n, or 10, 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. 514. It is also evident, that the logarithms of 1, 10, 100, 1000, etc. being 0, 1, 2, 3, respectively, the logarithm of any number, falling between 1 and 10, 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. 515. And, for a like reason, the logarithms of 1 1 1 etc. or of their equals, .1, .01, .001, 10' 100, 1000' etc. in the descending part of the scale, being- 1, -2, -3, etc. 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; and so on. 516. Hence, as the multiplying or dividing of any number by 10, 100, 1000, etc. 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 the same. quantity for the decimal part of their logarithms. 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. Ni+d; N = log. 10"xN=(i+m) +d; log. (i—m)+d; 10m where it is plain that the decimal part of the logarithm, in each of these cases, remains the same. 517. 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 use, we have 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. 518. Also, agreeably to what has been before observed, the logarithm of 38540 being 4.5859117, the logarithms of any other numbers, consisting of the same figures, will be as follows: Numbers. Logarithms. 3854 3.5859117 which logarithms, in this case, differ only in their indices, the decimal or positive part, being the same in them all. 519. And as the indices, or the integral parts of the logarithms of any numbers whatever, in this system, can always be thus readily found, from the simple consideration of the rule above-mentioned, they are generally omitted in the tables, being left to be supplied by the operator, as occasion requires. 520. 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 equations, a*=n, and en, which give (1) (2). x= log. n, x'= 1.n Where log. denotes the logarithm of n, in the system of which a is the base, and 1. its logarithm in the system of which e is the base. 521. Whence a*=e* or a*=e, and ea, we shall have, for the base a, log. e, and for the base e, = l.a; or (3). x' x' x=x' log. e, x'=x.l.a (4). Whence, if the values of x and x', in equations (1), (2), be substituted for x and x' in equations (3), (4), we shall have, log. n= log. exl.n, and l.n=- X 1 1 log.e log. n; or l.n=l.a × log. n, and log. n=; =l.n. where log. e, or its equal 1 1.a Ta expresses the constant Για ratio which the logarithms of n have to each other in the systems to which they belong. 522. 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 of which is 2.718281828459 523. Hence, in comparing this with the common or tabular logarithms, we shall have, by putting a in the latter of the above formula 10, the expression 1 = log. n= Xl.n, or l.n=1.10X log. n. 7.10 Where log., in this case, denotes the common logarithm of the number n, and l. its Neperian logarithm ; •4342944819 7.10 being what is usually called the modulus of the common or tabular system of logarithms. 524. It may not be improper to observe, that the logarithms of negative quantities, are imaginary; as has been clearly proved, by LACROIX, after the manner of EULER, in his Traité du Calcul Differentiel et Integral; and also, by SUREMAIN-MISSERY in his Théorie |