Purement Algébrique des Quantités Imaginaires. See, for farther details upon the properties and calculation of logarithms, GARNIER'S d'Algebre, or BonnyCASTLE'S Treatise on Algebra, in two vols. 8vo.

§ 11. APPLICATION OF LOGARITHMS TO THE SOLUTION

OF EXPONENTIAL EQUATIONS.

525. EXPONENTIAL EQUATIONS are such as contain quantities with unknown or variable indices: Thus, a=b, x=c, a=d, &c. are exponential equations.

y

526. An equation involving quantities of the form x*, where the root and the index are both variable, or unknown, seldom occur in practice, we shall only point out the method of solving equations involving

H

quantities of the form a, a, where the base a is constant or invariable.

527. It is proper to observe that an exponential of the form ab, means, a to the power of b*, and not a to the power of x.

Ex. 1. Find the value of x in the equation a=b. Taking the logarithm of the equation ab, we

have xx log. a=b;

log. b

; thus, let a=5,b=

Jog. a

100; then in the equation 5*=100,

log. 100_2.0000000

x=

log. 5

0.6989700

=2.864.

Ex. 2. It is required to find the value of x in the

equation a=c.

Assume by, then a=c, and y× log. a= log. c;

log. c

log. d

log. b the equation 93 = 1000,

log, a

the logarithm of the equation b=d, then, by (Ex. 1.),

Thus, let a 9, b=3, c=1000; then in

Hence b

log. (which let)=d. Take

C

log. d_log. 3.14___.4969296
log. 3

d); and a=

log. b

.4771213

=1.04.

Ex. 3. Make such a separation of the quantities in

x

the equation (a2 —b3)*=a+b, as to show that JC log. (a+b)

log. (a—b)

Taking the logarithm, we have

xx log. (a2 —b2)= log. (a+b), or x × log. (a+b)× (a-b)= log. (a+b);

that is, xx log. (a+b)+x × log. (a—b)= log. (a+b). Hence x x log. (a—b)=log. (a+b)—x × log. (a+b) x log. (a+b) =(1−x) log. (a+b); •°• =

-xlog. (a-b)*

Ex. 4. Given a+b=c, and a*—by=d, required the values of x and y.

By addition, 2a*=c+d, or a2=c+d, which p

then = log. a

a*:

put =

Again, by subtraction, we have 26=c-d, or ¿3=

-d

2

(which letn) ; •'• y=

log. n

log. b

Ex. 5. Find the value of x in the equation

ab®+c=e. d

log. (de-c)-log. a

Ans. x=

log. b

Ex. 6. Find the value of x in the equation a

√(b2 —c2) Vd3e

Ans. x=

log. (b+c)+ log. (b—c)- log. d-log.e

log. a

Ex. 7. Find the value of x in the equation a* +fo

Ex. 8. Given log. x+ log. y= 1⁄2) to find the vaand log. x-log. y=lues of x and y. Ans. x=10/10 and y=10. Ex. 9. In the equation 3o=10, it is required to find the value of x. Ans. x=3.32198, &c. Ex. 10. Given /729=3, required the value of x. Ans. x=6. Ex. 11. Given 57862-8, to find the value of x. Ans. 5.2734, &c.

Ex. 12. Given (216)=64, to find the value of x. Ans. x 4.2098, &c.

Ex. 13. Given 43=4096, to find the value of x.

log. 3

Ex. 14. Given a*+y=c, and bad, to find the va

THE RESOLUTION OF EQUATIONS
OF THE THIRD AND HIGHER DEGREES.

§ I. THEORY AND TRANSFORMATION OF EQUATIONS.

528. In addition to what has been already said (Art. 192), it may here be observed, that the roots of any equation are the numbers, which, when substituted for the unknown quantity, will make both sides of the equation identically equal. Or, which is the same, the roots of any equation are the numbers, which, substituted for the unknown quantity, reduce

the first member to zero, or the proposed equation to the form of 0=0; because every equation may, designating the highest power of the unknown quantity by (x), be exhibited under the form

x+Axm-1+Bx2+Сx-3+.. Tx+V=0.(1), A, B, C,.... T, V, being known quantities. And the resolution of an equation is the method of finding all the roots, which will answer the required condi

tion.

529. This being premised, it may now be shown, that if a be a root of the equation (1), the left-hand member of that equation will be exactly divisible by

For if a be substituted for x, agreeably to the above definition, we shall necessarily have a+AaTM-1+Bam--3+Cam--3+... Ta+V=0. And consequently, by transposition,

V=-am-Aam--1-Bam--2—Cam--3 —

...

-

- Ta.

Whence, if this expression be substituted for V in the first equation, we shall have, by uniting the corresponding terms, and placing them all in a line, (xTM—-aTM)+A

(x2--1—am-1)+B(xm--2—uTM--2)+T(x−a)

=0.

Where, since the difference of any two equal powers of two different quantities is divisible by the difference of their roots (Art. 108), each of the quantities (-a), (xm--1 — am-1), (xm--2-am-2), &c. will be divisible by x-Ar And, therefore, the whole compound expression

(x-am)+A(xm-am-1)+B(x2-am-2)+
&c.=0,

which is equivalent to the equation first proposed, is also divisible by x-a; as was to be shown.

But if a be a quantity greater or less than the root, this conclusion will not take place; because, in that case, we shall not have

1

V=—am — Aam--ı — Bɑm~2 —Сam--3 . -Ta; which is an equality obviously essential to the division in question.

530. The preceding proposition may be demonstrated, after the manner of D'ALEMBERT, as follows: In fact, designating by X, the polynomial, which forms the first member of the equation (1); then we shall always carry on the division of X by x-a, till we arrive at a remainder R, independent of x, since x is only of the first degree in the divisor; so that, representing by Q the corresponding quotient, we shall have this identity,

X=Q(x-a)+R.

Now, by hypothesis, a substituted for a reduces the polynomial X to zero; and it is evident that the same substitution gives Q(x-a)=0; therefore we shall necessarily have 0=R: Hence -a divides the equation (1), without a remainder.

Reciprocally, if the first member of any equation of the form X=0 be divisible by x-a, a is a root. In fact we have, according to this hypothesis, the identity X=Q(x-a), which, for x=0, gives X=0; therefore, (Art. 528), a is a root of the proposed equation.

COR. I. Hence we may easily conclude, that if a be not a root of the equation (1), the first member will not be divisible by x-a.

COR. 2. And if the first member of the equation (1), be not divisible by x-a, a is not a root of the proposed equation.

531. Supposing every equation to have one root, or value of the unknown quantity, it can then be shown, that any proposed equation will have as many roots as there are units in the index of its highest term, and no more. For let a, according to the assumption here mentioned, be a root of the equation (1),

x+Ax+B2-3 +Cx-3+...+Tx+V=0. Then, since by the last proposition this is divisible by x-a, it will necessarily be reduced, by actually performing the operation, to an equation of the next inferior degree, or one of the former

x-1+Ã2x2+B'x-3+Cx-+.. T'x+V'=0