381. The same rule will also apply when the binomial contains impossible or imaginary quantities. Let it be proposed, for example, to extract the root of 1+4√3.

382. Again, let it be required to extract the root of 2 √✓ — 1. As we have here no rational part at all, we have,

b=-4.

.•. a2—b=c2=4 and c=2. .'./a+c

-c

2

2=√1+√=I=1+√/−1.

the square of which is 1+2-1-1=2√-1.

EXAMPLES.

Ex. 1. Extract the square root of 2+2 √✓−3.

a=2

√6=2√3

b=-12

2

+

.. a2bc2 16, and c=4. ...
:: Va+c+

Vate

2

Ex. 2. Extract the square root of 8+ √15.

Ex. 3. Reduce 16 to a binomial of the form of /x+

√y.

Since -16= √16× √-1=4√−1;

aa—b=c2=16, c=4;. ̧.·.\ /a+c+==

Ex. 10. Find the values of x, in the equation x-6x2+ 4=0?

Ex. 11. Find the values of x, in the equation — 12x2+ 16=0?

ANS. 51.

Ex. 12. Required two numbers, such that their sum, their product, and the difference of their squares, may be all equal?

Ex. 13. Required two numbers, such, that their sum, their product, and the sum of their squares, may be all equal?

383. IN

CHAP. XLVIII.

OF LOGARITHMS IN GENERAL.

any equation of the form of ab, we have already seen that a is called the root, x the exponent, and a the power. We have also seen how the value of b may be determined, when the root a and the exponent x are known. For instance, if a 5 and x=3, by the substitution of these values in the equation, we determine b=53125. We have also seen how, when b and ≈ are known, the value of a might be determined; as, if b=125 and 3, we know at once that a=/125=5. But we have not yet seen how the exponent x is to be found, the root a and the representative value of the power a being known.

384. On this question is founded the important theory of logarithms; the use of which is so extensive through the whole compass of mathematics, that scarcely any long calculation can be carried on without their assistance.

385. Resuming then the equation ar=b, we shall begin by remarking that, in the doctrine of logarithms, we assume for the root a, a certain number taken at pleasure, and suppose this root to preserve invariably its assumed value. This being laid down, we take the exponent x such, that the power a becomes equal to a given number b; in which case this ex

ponent

ponent a is said to be the logarithm of the number b, and is thus expressed, x = log. b.

386. From hence it appears, that the value of the root a being once determined, the logarithm of any number b is nothing more than the exponent of that power of a which is equal to b; so that b being=a*, x is the logarithm of the power a*.

387. Now, if we suppose x = 1, we shall have 1 for the logarithm of a1, and consequently log. a = 1. If x=2, we have 2 for the logarithm of a2, and log. a2 = 2,

and so on.

388. Again, if we make x = 0, then will O be the logarithm of ao. But we have seen that ao= 1; and, consequently, log. 10, whatever be the value of the root a.

Further, if x=-1, then-1 will be the logarithm of

1

1

a-1. But a-1 = ; so that we have log. -=

α

same manner, we shall have log.

so on.

389. We have thus seen how the logarithms of all the powers of a may be represented, and even those of fractions, which have unity for the numerator, and for the denominator any power of a. We likewise see that in all those cases the logarithms are integers; but it is necessary to observe that, if x were a fraction, it would be the logarithm of an irrational number: if, for instance, we suppose x, it follows is the logarithm of a3, ora. Consequently, we have log. √a={}, and in the same manner, we shall find log. a=; log. a, &c.

390. If, however, it be required to find the logarithm of another number b, it will readily be perceived that it can nei

ther be an integer, nor a fraction; yet there must be such an exponent x, that the power a* may become equal to the number proposed, or that x= log. b, and, generally, that a log.bb.

391. Let us now consider another number c, whose logarithm has been represented in a similar manner by log. c, so that a log. cc. If we multiply this equation by the preceding one, we shall have a log. 6+ log. c = bc: hence the exponent is always the logarithm of the power, and we have therefore log. b + log.cbc.

392. If, on the other hand, we had divided the first equation by the second, we should have had a log.blog.c

393. Thus have we arrived at the two principal properties of logarithms, which are shown in the equations log. b + log. cbc, and log. blog. c = The former of these equa

b

C

tions teaches us that the logarithm of a product, as bc, is found by adding together the logarithms of its factors; and the latter, that the logarithm of a fraction may be found by subtracting the logarithm of the denominator from the logarithm of the numerator; and it follows from hence, that two numbers may be multiplied or divided, by the addition' or subtraction of their logarithms; and thus we perceive the singular utility of logarithms in extensive and complicated calculations.

394. Moreover, logarithms are attended with even greater advantage in the involution of powers and the extraction of roots: for, supposing bc, we have by the first property log. c + log, c=c2, or 2 log. c = log. c2; and in like manner 3 log. c = log. c3; 4 log. c = log. c, and generally log.c = log. c". If therefore n be a fraction, we shall have

log.