CHAPTER XIV. ON LOGARITHMS. 495. Previous to the investigation of Logarithms, it may not be improper to premise the two following propositions. 496. Any quantity which from positive becomes negative, and reciprocally, passes through zero, or infinity. In fact, in order that m, which is supposed to be the greater of the two quantities m and n, becomes n, it must pass through n; that is to say, the difference m―n becomes nothing; therefore P, being this difference, must necessarily pass through zero, in order to become negative or-p. But if p becomes —p, the fraction will become; and therefore it passes through, or infinity. 497. It may be observed, that in Logarithms, and in some trigonometrical lines, the passage from positive to negative is made through zero; for others of these lines, the transition takes place through infinity: It is only in the first case that we may regard negative numbers as less than zero; whence there results, that the greater any number or quantity a is, when taken positively, the less is -a; and also, that any negative number is, a fortiori, less than any absolute or positive number whatever. 498. If we add successively different negative quantities to the same positive magnitude, the results shall be so much less according as the negative quantity becomes greater, abstracting from its sign. For instance, 8-1>8—2 、 8—3, &c. It is in this sense, that 0> -1>—2>-3, &c.; and 3> 0>-1>—2—3>—4, &c. 499. Any quantity, which from real becomes imaginary, or reciprocally, passes through zero, or infinity. This is what may easily be concluded from these expressions, x=√(a2—y2), x= 1 ✔(a2 - y2) considered in these three relations, y' La2, y=a2, y2、a2. § I. THEORY OF LOGARITHMS. 500. LOGARITHMS are a set of numbers, which have been computed and formed into tables, for the purpose of facilitating arithmetical calculations; being so contrived, that the addition and subtraction of them answer to the multiplication and division of the natural numbers, with which they are made to correspond. 501. Or, when taken in a similar, but more general sense, logarithms may be considered as the exponents of the powers, to which a given, or invariable number, must be raised, in order to produce all the common, or natural numbers. Thus, if ay, ax=y', ax" =y", &c. ; then will the indices x, x', x", &c, of the several powers of a, be the logarithms of the numbers y, y', y", &c. in the scale or system, of which a is the base. 502. So that, from either of these formulæ, it appears, that the logarithm of any number, taken separately, is the index of that power of some other number, which, when it is involved in the usual way, is equal to the given number. And since the base a, in the above expressions, can be assumed of any value, greater or less than 1, it is plain that there may be an endless variety of systems of logarithms, answering to the same natural numbers. 503. Let us suppose, in the equation ay, at first, x=0, we shall have y=1, since (Art. 453), ao=1; to x=1, corresponds y=a. Therefore, in every system, the logarithm of unity is zero; and also, the base is the number whose proper logarithm, in the system to which it belongs, is unity. These properties belong essentially to all systems of logarithms. 504. Let +x be changed into -x in the above equation, and we shall have 1 =y: ax Now, the exponent x augmenting continually, the fraction , if the base a be greater than unity, will diminish, and may be made to approach continually towards 0, as its limit; to this limit corresponds a value of x greater than any assignable number whatever. Hence it follows, that, when the base a is greater than unity, the logarithm of zero is infinitely negative. 505. Let y and y' be the representatives of two numbers, x and x' the corresponding logarithms for the same base: we shall have these two equations, a*=y, and a*'-y', whose product is a.ay.y', or ax+x=yy', and consequently, by the definition of logarithms, (Art. 501), x+x'=log. yy', or log. yy' log. y-log. y. And, for a like reason, if any number of the equations a=y, a*=y', ax"=y", &c. be multiplied together, we shall have a+++etc.yyy", &c. ; and, consequently, x+x'+x", &c. log, yyy", &c.; or log. yyy", &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 a3=y, be divided by ax=y', we shall have, from the nature of powers, X-X' y ax ax or a ; and by the definition of logarithms, x-x'=log. (24) ; or log. y—log. y'= 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 rais ed to the fractional power m consequently, as before, x=log. (y")=log. ym; or log. 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 I 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 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. 512. So that, according to this scale, which is that of the common logarithmic tables, the numbers, -4 -3 --2 3 4 etc. 10 10 10 10 10, 10, 10 10, 10, 1 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 10n, 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 10' 100 1 1000 etc. or of their equals, .1, .01, .001, etc. in the des cending 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; log. 10m XN N 10m =(i+m)+d; log. (im)+d; 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 consias, 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: |
