1. Required the logarithm of 1.07632.

This number is found between 1.076 and 1.077; hence

2. Required the logarithm of 3579.

In order to make use of table II, we proceed thus: 3579 ÷ 35 = 102.25714+.

log. 102.3 log. 102.2425;

log. 102.22.009451

n =.5714.

NOTE.-It is obvious that if we divide any number by its first two figures,

we may obtain the logarithm of the quotient by means of table II; then wo may add the logarithm of the divisor, found by table I, to obtain the required logarithm.

EXPONENTIAL EQUATIONS.

418. We will now illustrate the application of logarithms to the solution of exponential equations.

1. Given 210 to find the value of x.

Suppose the logarithms of both members of the equation to be taken. We shall have, by 404, 5,

Raising both members of the given equation to the power denoted by x, we have

Taking the logarithms of both members,

log. 25 x log. 3x log. 7;

whence,

3. Given rax = bc to find the value of x.

Taking the logarithms of both members of the equation, we have, by 404, 3 and 5,

SECTION VIII.

PROPERTIES OF EQUATIONS.

419. Let us assume the equation,

x+Axm-1+ Bxm-2 + .... + Tx + U = 0 ... (1),

in which m, the exponent of the degree, is a positive whole number. An equation not given in this form may be readily reduced to it, by transposing all the terms to the first member, arranging them according to the descending powers of the unknown quantity, and dividing through by the coefficient of the first term.

In this equation the coefficients, A, B, C, etc., may denote any quantities whatever; that is, they may be positive or negative, entire or fractional, rational or irrational, real or imaginary. The term U may be regarded as the coefficient of 2o, and is called the absolute term of the equation.

420. If the equation contains all the entire powers of x, from the mth down to the zero power, it is said to be complete; if some of the intermediate powers of x are wanting, it is said to be incomplete. An incomplete equation may be made to take the form of a complete equation, by writing the absent powers of x with 0 for their coefficient.

421. It has been shown (305) that any expression of the second degree containing but one unknown quantity, may be resolved into two binomial factors of the first degree with respect to the unknown quantity,-the first term in each factor being this quantity, and the second term one of the roots (with its sign changed) of the equation which results from placing the expression equal to zero. We therefore conclude that every expression of the second degree may be regarded as the product of two binomial factors of the first degree.

So likewise the product of three binomial factors of the first degree with respect to any unknown quantity, will be an expres

sion of the third degree, and we readily see that by varying the values of the second terms of the factors, corresponding changes are produced in the product. Thus,

(x − 2) (x+3)(x — 5) = x3 — 4x2 - 11x+30,

(x − 2 + √−3) (x − 2 − √ − 3) (x + 4) = x3 − £x2 + 42, (x + 1 − √− 3) (x + 1 + √ − 3) (x − 2) — x3 — 8.

=

From these and other examples, which may be increased at pleasure, it is inferred that any expression of the third degree in respect to x, would result from the multiplication of some three factors of the first degree in respect to x. And in general, any expression of the mth degree with respect to its unknown quantity, may be regarded as the result of the multiplication of m binomial factors of the first degree with respect to that unknown quantity.

422. If then we have any equation formed by placing a polynomial containing the unknown quantity, x, equal to zero, and we discover the binomial factor x a in the first member, it is evident that a is a root of the equation; for, when substituted for x, it reduces the first member to zero.

If we can succeed, therefore, in discovering the binomial factors of the first degree, of the first member of any equation, the roots of the equation will be the values of x obtained by placing each of these factors, successively, equal to zero.

This reverse process of resolving the first member of an equation into its binomial factors of the first degree, is one the difficulty of which increases rapidly with the degree of the equation; and algebraists have as yet discovered no general method for effecting this resolution for those of a higher degree than the fourth. By special processes, however, the roots of numerical equations may be found exactly, when commensurable, and to any degree of approximation when not commensurable.

of

423. In order to discover the law which governs the product any number of binomial factors, such as x + a, x + b, x + c, etc., having the first term the same in all, and the second terms different, let us first obtain the product of several of these factors by actual multiplication; thus,