mention the tables of Callet, stereotype edition, and those of Borda, as very complete and very convenient. 250. If we have the logarithm of a number y for a particular value of a, or for a particular base, it is easy to obtain the logarithm of the same number in any other system. If we have a = y; for another base A, we have A = y, X being different from x; hence we deduce A= a*. Taking the logarithms according to the system the base of which is a, we have 1 A = la* ; now lax by hypothesis, and 1A3 = X1A (241); therefore, X1A = x, or X=; but if we consider A as a base, X will be 1A the logarithm of y in the system founded on this base; if, therefore, we designate this last by Ly, in order to distinguish it from the other, we have and we find the logarithm of y in the second system, by dividing its logarithm taken in the first by the logarithm of the base of the second system. The preceding equation gives also ly Ly 1A; from which it is evident, that whatever be the number y, there is between the logarithms ly and Ly, a ratio invariably represented by 1A. 251. In every system the logarithm of 1 is always 0, since whatever be the value of a we have always a° = 1. As logarithms may go on increasing indefinitely, they are said to become infinite at the same time with the corresponding numbers; and as, when y is a fractional number, we have y = = = a*, it 1 a* is evident, that in proportion as y becomes smaller, x in its negative state becomes greater, but we can never assign for x a number, which shall render y strictly nothing. In this sense it is said, that the logarithm of zero is equal to an infinite negative quantity, as we find in many tables. 252. I now proceed to give some examples of the use, which may be made of logarithms in finding the numerical value of formulas. It follows from what is said in art. 241, and from the definition of logarithms, by which we are furnished with the equation aly = y, that which is very complicated, we find 1(A2 / B2 — C2) = 1[A2 √√(B+ C) (BC)] 5 1 (C √√/ D3 E F) = 1 C + ¿ 1 D + † 1 E + † 1F, and, consequently, = 21 A + 1 (B+ C) + } 1 (B — C) —1 C—}\D—} \E— }\F. ¦ If we take the arithmetical complements of 1 C, 31D, ¦ \ E, ¦ \ F, designating them by C', D', E', F', instead of the preceding result, we have 2 1 A + ¦ 1 (B + C) + ¦ 1 (B − C) + C' + D' + E' + F', only we must observe to subtract from the sum as many units of the same order with the complements, as there are complements taken, that is 4. When we have found the logarithm of the proposed formula, the tables will show the number, to which this logarithm belongs, which will be the value sought. 253. Logarithms are of most frequent use in finding the fourth term of a proportion. It is evident, that if a: b:: cd we have =bc, whence 1d=1b + 1 c—la; d a that is, the logarithm of the fourth term sought is equal to the sum of the logarithms of the two means, diminished by the logarithm of the known extreme, or rather, to the sum of the logarithms of the means, plus the arithmetical complement of the logarithm of the known extreme. 254. If we take the logarithms of each member of the equab d which presents the character of a proportion, we tion have a = lb-lald-1c (252); 1b-lalc-1b1d-lcle-1d, &c., and hence we infer, that the progression by quotients, abcd: e, &c. corresponds to the progression by differences, la.lb.lc.ld.le, &c., and. consequently, the logarithms of numbers in progression by quotients, form a progression by differences. 255. If we have the equation b = c, we may easily resolve it by means of logarithms; for as 1 b is equal to z lb, we have z 1b1c, and, consequently, z = The equation b = d may be resolved in the same manner; making c=u, we have 1 c 16 In this last expression, 11b represents the logarithm of the logarithm of b, and is found by considering this logarithm as a num ber. The quantities, , le, and all which are derived from t≈, them, are called exponential quantities. Questions relating to the Interest of Money. 256. THE principles of progression by quotients and of logarithms will be found to occur in the calculations relating to interest. To understand what I have to offer on this subject it must be recollected, that the income derived from a sum of money employed in trade, or in executing some productive work will be in proportion to the frequency with which it is exchanged in either case. Hence it follows, that he, who borrows a sum of money for any purpose, ought, upon returning this money at the expiration of a given time, to allow the lender a premium equivalent to the profits, which he might have received, if he had employed it himself. Such is the view in which the subject of interest presents itself. In order to determine the interest of any sum, we compare this sum with 100 dollars taken as unity, having fixed the premium, which ought to be allowed for this last at the end of a particular term, one year for example. I shall not here consider those things, which in the different kinds of speculation, occasion the rise and fall of interest; this belongs to the elements of political and commercial arithmetic, which should be preceded by some account of the doctrine of chances. My object in what follows is simply to resolve certain questions, which refer themselves to progression by quotients. To present the subject in a general point of view, I shall suppose the annual premium, allowed for a sum 1, to be represented by r, r being a fraction; it is evident, that the interest of a sum 100, for the same time, will be 100 r, that of any sum whatever a will be denoted by a r; if we designate this last by a, we have α = ar. By means of this formula, it is easy to find the interest of any sum whatever, when that of 100 or of any other sum, for a known time, is given; questions of this kind belong to what is called simple interest. 257. But if the lender, instead of receiving annually the interest of his money, leaves it in the hands of the borrower to accumulate, together with the original sum, during the following year, the value of the whole at the end of this year may be found in the following manner. The original sum being a, if we add to it the interest a r, it becomes at the end of the first year Now if we make a + ar = a(1 + r), a (1 + r) = a', the interest of the sum a' for one year being a' r, that of the sum a(1+r) will be, for a second year, ar (1 + r); and as, at the end of the first year, the principal a, augmented by the interest, becomes a (1+r), the principal a amounts, at the end of the second year, to Alg. a' (1 + r) a (1 + r)2 = a'. = 33 If the lender does not now withdraw the sum a", but leaves it to accumulate during a third year, at the end of this, it will become, according to what precedes, a" (1 + r) = a (1 + r)3 = a'". It will be readily perceived, that a"" will become at the end of the fourth year a''' (1 + r) = a (1 + r)a, and so on; and that, consequently, the sum first lent, and the several sums due at the end of the first, second, third, fourth, &c. years, form the following progression by quotients; aa (1+r) a (1 + r)2 : a (1 + r) 3 : a (1 + r)1 : &c. of which the quotient is 1+r, and the general term a (1 + r)” = A, the number n representing the number of years, during which the interest is suffered to accumulate. If the rate of interest be 5 per cent., for example, that is, if for 100 dollars during one year 105 dollars are paid back; we have 100 r = 5, or r = 150 =, and 1 + r = }}}· If we would know to what the sum a amounts, when left to accumulate during 25 years, we have instead of the original sum. The 25th power of may be easily found by means of logarithms, since we have (252) 25 = 2511 = 25 (121 120) = 0,5297322, 1 (21) " == 20 which gives 21 25 3,386 nearly, A = 3,386 a ; and hence it may be readily seen, that 1000 dollars will in this way amount at compound interest to 3386 dollars, at the end of 25 years. If the sum lent were for 100 years, we should have nearly; thus 1000 dollars would produce, at the end of this period, a sum of 131000 dollars nearly. These examples will be |