Elements of Algebra

than that of the denominator, and, consequently, if we subtract the latter from the former, the result will be negative.

In order to obtain the logarithm of the fraction, for example, we subtract from 0, which denotes the logarithms of 1, the fraction 0,3010300, which represents that of 2; the result is

If we subtract from 0 the number 1,3010300, which is the logarithm of 20, we have the logarithm of, equal to

-1,3010300.

The logarithm of 3 being 0,4771213, that of will be

0,3010300-0,4771213-0,1760913.

248. It is evident from the manner in which the logarithms of fractions are obtained, that, considered independently of their signs, they belong (241) to the quotients, arising from the division of the denominator by the numerator, and, consequently, answer to the number by which it is necessary to divide unity in order to obtain the proposed fraction. Indeed, for example, may be exhibited under the form, and 1 = 13-12= 0,1760913.

It would be inconvenient, in order to find the value of a fraction, to which a given negative logarithm belongs, to employ the number to which the same logarithm answers when positive, since it would be necessary to divide unity by this number; but if we subtract this logarithm from 1, 2, 3, &c. units, the remainder will be the logarithm of a number, which expresses the fraction sought, when reduced to decimals, since this subtraction answers to the division of the numbers, 10, 100, 1000, &c. by the number to which the poposed logarithm belongs.

Let there be, for example, -0,3010300; if, without regarding the sign, we take this logarithm from 1, or 1,0000000, the remainder, 0,6989700, being the logarithm of 5, shows, that the fraction sought is equal to 0,5, since we supposed unity to be composed of 10 parts.

If, in seeking the logarithm of a fraction, we conceive unity to be made up of 10, or 100, or 1000, &c. parts, or which amounts to the same thing, if we augment the characteristic of the logarithm of the numerator by a number of units sufficient to enable us to subtract that of the denominator from it, we obtain in this way a positive logarithm, which may be employed in the place of that indicated above.

In order to introduce uniformity into our calculations, we most frequently augment the characteristic of the logarithm of the numerator by 10 units. If we do this with respect to the fraction, for example, we have

It will be readily seen, that this logarithm exceeds the negative logarithm— 0,1760913 by 10 units, and that, consequently, whenever we add it to others, we introduce 10 units too much into the result; but the subtraction of these ten units is easily performed, and by performing it we effect at the same time the subtraction of 0,1760913. Let N be the number, to which we add the positive logarithm 9,8239087; the result of the operation will be represented by

100,1760913. and if we subtract 10, we have simply

N-0,1760913.

According to the preceding observations, we cause addition to take the place of subtraction, by employing, instead of the number to be subtracted, its arithmetical complement, that is, what remains, when this number is subtracted from one of the numbers, 10, 100, 1000, &c., a result which is obtained by taking the units of the proposed number from 10 and the several other figures from 9. We add this complement to the number, from which the proposed logarithm is to be subtracted, and from the sun subtract an unit of the same order as the complement.

It is evident, that if the complement is repeated several times, we must subtract, after the addition, as many units of the same order with the complement, as there are in the number, by which it is multiplied; and, for the same reason, if several complements are employed, we must subtract for each an unit of the same order, or as many units as there are complements, if they are all of the same order.

Sometimes this subtraction cannot be effected; in this case, the result is the arithmetical complement of the logarithm of a fraction, and answers in the tables to the expression of this fraction reduced to decimals. If 10 units remain to be taken from the characteristic, as is most frequently the case, the result is the same as if we had multiplied by 10000000000, the numerator of the fraction. sought, in order to render it divisible by the denominator; the char

acteristic of the logarithm of the quotient shows the highest order of the units contained in this quotient, considered with reference to those of the dividend. In 9,8239087, the characteristic 9 shows, that the quotient must have one figure less than the number, by which we have multiplied unity; and, consequently, if we separate 10 figures for decimals, the first significant figure on the left will be tenths; and we shall find only hundredths, thousandths, &c., for the numbers the arithmetical complements of which have 8, 7, &c. for their characteristics.

249. What has been said respecting the system of logarithms, in which a 10, brings into view the general principles necessary for understanding the nature of the tables; for more particular information the learner is referred to the tables themselves, which usually contain the requisite instruction relating to their arrangement and the method of using them. I will merely 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 Ax = 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

1A2 = 1a*;

now lax by hypothesis, and 1 A* = X1A (241); therefore,

X1A = x, or X = ; but if we consider ♬ as a base, X will

ΙΑ

be 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

=1A; from which it is

evident, that whatever be the number y, there is between the loga

rithms 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 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.

απ

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 a1y = y, that

1(AB) =1A+1B, 1(1)=14—1B,

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1 (A2 √√ B2 — C2) = 1 [Æ2 √√(B + C) (B — C)] =

· =

1(C√ √ D3 EF) = 1 C + { ID + } 1 E + } 1F,

21A + 1 (B+ C) + 11 (B-C) -1C-1D-11E-}|F, If we take the arithmetical complements of 1 C, 1D, }]E, ¦|F, designating them by C', D', E', F', instead of the preceding result, we have

21A + 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 the most frequent use in finding the fourth term of a proportion. It is evident, that if a : b::c: d we have

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 equation

b d

2, which presents the character of a proportion, we have

1b-lald-lc (252);

whence it follows, that the four logarithms