And, by Art. 195, the general term of the expansion of

where ẞ, Y, 8 ... are positive integers whose sum is p.

Hence the general term in the expansion of the given expression is

197. Since (a + bx + cx2 + dx3 +...)" may be written in the form

it will be sufficient to consider the case in which the first term of the multinomial is unity.

Example. Find the coefficient of x3 in the expansion of

We have to obtain by trial all the positive integral values of ß, y, d which satisfy the equation B+2y+38=3; and then p is found from the equation p=8+y+d. The required coefficient will be the sum of the corresponding values of the expression (1).

In finding ß, y, d, ... it will be best to commence by giving to d successive integral values beginning with the greatest admissible. In the present case the values are found to be

d=1, y=0, B=0, p=1;

d=0, y=1, B=1, p=2;

d=0, y=0, B=3, p=3.

Substituting these values in (1) the required coefficient

198. Sometimes it is more expeditious to use the Binomial Theorem.

Example. Find the coefficient of x4 in the expansion of (1 − 2x+3x2)−3. The required coefficient is found by picking out the coefficient of x1 from the first few terms of the expansion of (1-2x-3x2)- by the Binomial Theorem; that is, from

1+3 (2x-3x2)+6 (2x – 3x2)2 + 10 (2x – 3x2)3 +15 (2x – 3x2)4;

we stop at this term for all the other terms involve powers of x higher than x4.

The required coefficient=6.9+10. 3 (2)2 (− 3) +15 (2)1

=-66.

EXAMPLES. XV.

Find the coefficient of

1. a2b3c4d in the expansion of (a-b-c+d)10.
2. a2b5d in the expansion of (a+b-c-d)3.
3. a3b3c in the expansion of (2a+b+3c).
4. x2y3 in the expansion of (ax−by+cz)9.
5. 23 in the expansion of (1+3x-2.x2)3.
6. 24 in the expansion of (1+2x+3x2)1o.

7. 2 in the expansion of (1+2x − x2)5.
8. 8 in the expansion of (1-2x+3x2 - 4x3)4.

Find the coefficient of

9. x23 in the expansion of (1 − 2x+3x2 − x1 — 25)5.

13. 24 in the expansion of (2-− 4x+3x2) −2.
14. 26 in the expansion of (1+4x2+10x+20x6)
15. 12 in the expansion of (3−15x3+18x6)−1.

1

(2) α12+2α2+3α+......+np. amp = 1 np (p + 1)".

20. If αo, a1, a2, az... are the coefficients in order of the expansion of (1+x+x2)", prove that

az2+.

-1 2

a ̧2 — a ̧3 + a‚ ̧3 — a ̧3 +........... + ( − 1)" − 1an - 1 = '1' a„ {1 − ( − 1)”an}.

21. If the expansion of (1+x+x2)μ

CHAPTER XVI.

LOGARITHMS.

199. DEFINITION. The logarithm of any number to a given base is the index of the power to which the base must be raised in order to equal the given number. Thus if a = N, x is called the logarithm of N to the base a.

Examples. (1) Since 34-81, the logarithm of 81 to base 3 is 4.

(2) Since 101=10, 102-100, 103-1000,......

the natural numbers 1, 2, 3,... are respectively the logarithms of 10, 100, 1000,...... to base 10.

200. The logarithm of N to base a is usually written log. N, so that the same meaning is expressed by the two equations

Example. Find the logarithm of 32 5/4 to base 2/2.

Let x be the required logarithm; then,

201. When it is understood that a particular system of logarithms is in use, the suffix denoting the base is omitted. Thus in arithmetical calculations in which 10 is the base, we usually write log 2, log 3,...... instead of log,,2, log1,3,.......

10

10

Any number might be taken as the base of logarithms, and corresponding to any such base a system of logarithms of all numbers could be found. But before discussing the logarithmic systems commonly used, we shall prove some general propositions which are true for all logarithms independently of any particular

base.

202. The logarithm of 1 is 0.

For a

the base

H

1 for all values of a; therefore log 10, whatever may be.

203. The logarithm of the base itself is 1.

For a' a; therefore log,a= 1.

204.

=

To find the logarithm of a product.

Let MN be the product; let a be the base of the system, and suppose

Similarly, log, MNP = log1M + log, N + log1P;

and so on for any number of factors.