EXAMPLES XXXVII. b.

In the following expansions find which is the greatest term:

1. (x + y)17 when x = 4, y = 3.

2. (xy)28 when x = 9, y = 4. 3. (1+x) when x = ૐ.

4. (a-4b)15 when a = 12, b = 2

5.

(7x+2y)30 when x=8, y=14. 6. (2x+3)" when x = 1⁄2, n = 15.

7. In the expansion of (1 + x)25 the coefficients of the (2 r + 1)th and (r+5)th terms are equal: find r.

8. Find n when the coefficients of the 16th and 26th terms of (1+x)" are equal.

9. Find the relation between r and n in order that the coefficients of (r+3)th and (2r — 3)th terms of (1 + x)3n may be equal.

10. Find the coefficient of xm in the expansion of (x2 +

2n

(x2 + 1)2m.

11. Find the middle term of (1 + x)2n in its simplest form.

12. Find the sum of the coefficients of (x + y)16.

13. Find the sum of the coefficients of (3x + y)o.

14. Find the rth term from the beginning and the rth term from the end of (a + 2 x)”.

15. Expand (a2 + 2 a + 1)3 and (x2 - 4x + 2)3.

+ x)=.

Write in simplest form :

25. The 5th term and the 10th term of (1 + 2

26. The 3d term and the 11th term of (1 + 2x)113.

27. The 4th term and the (r + 1)th term of (1 + x)−2.

28. The 7th term and the (r + 1)th term of (1

29. The (r + 1)th term of (a — bx)-1, and of (1 — nx)".

Find to four places of decimals the value of

30. 122.

31. 620.

32.

31.

33. 1÷√99

Find the value of

84. (x +√2)4 + (x − √2)1.

36. (√2 + 1)6 − ( √2 − 1)o.

35. (√x2-a2+x)3 — (√x2-a2—x)3. 37. (2−√1−x)+(2+√1−x)°.

16

49. The (r+1)th term of (1−x)−4.

50. The (r+1)th term of (1+x)*. 51. The (r+1)th term of (1+x)1

47. The 5th term of (3 a 2b)-1. 52. The 14th term of (210-27 x) * 48. The (r+1)th term of (1-x)-2. 53. The 7th term of (38+6x)*.

CHAPTER XXXVIII.

LOGARITHMS.

425. 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, 108 = = 1000, are respectively the logarithms of 10,

the natural numbers 1, 2, 3, 100, 1000, to base 10.

...

...

...

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

a=N; x=log. N.

Ex. Find the logarithm of 32/4 to base 2√2.

Let x be the required logarithm; then, by definition,

(2√2) = 32/4; ••. (2 • 23)* = 25 . 23 ; 2* = 25+3;

...

hence, by equating the indices, x = 27;

... x = 18 = 3.6.

427. 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 log10 2, log10 3,....

Logarithms to the base 10 are known as Common Logarithms; this system was first introduced in 1615 by Briggs, a contemporary of Napier the inventor of Logarithms.

PROPERTIES OF LOGARITHMS.

428. Logarithm of Unity. The logarithm of 1 is 0.

For ao

=

1 for all values of a; therefore log 1 = 0, whatever the base may be.

429. Logarithm of the Base. The logarithm of the base itself is 1.

For a1a; therefore log, a = 1.

430. Logarithm of Zero. The logarithm of 0, in any system whose base is greater than unity, is minus infinity.

Also, since a+= ∞, the logarithm of +∞o is +∞o.

431. Logarithm of a Product. The logarithm of a product is the sum of the logarithms of its factors.

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

so that

x = log. M; y = log. N;

Thus the product MN= a* xa" = a*+";

whence, by definition,

Similarly,

log, MN=x+y=log, M+log, N.

log, MNP = log. M+log, N+log. P;

and so on for any number of factors.

Ex.

log 42 = log (2 × 3 × 7) = log 2 + log 3 + log 7.

432. Logarithm of a Quotient. The logarithm of a quotient is the logarithm of the dividend minus the logarithm of the divisor.

433. Logarithm of a Power. The logarithm of a number raised to any power, integral or fractional, is the logarithm of the number multiplied by the index of the power.

Let log. (M2) be required, and suppose

Ex. Express the logarithm of as in terms of log a, log b, and log c.

at

c5b2

= log = log a log (c5b2)

c5b2

loga - (log c5 + log b2) = & log a - 5 log c - 2 log b.

=

434. From the equation 10 N, it is evident that common logarithms will not in general be integral, and that they will not always be positive.

For instance,

Again,

3154 103 and <10*;

.. log 31543+ a fraction.

.06 10-2 and < 10-1;

... log.062+ a fraction.

Negative numbers have no common logarithms.

435. DEFINITION. The integral part of a logarithm is called the characteristic, and the decimal part, when it is so written that it is positive, is called the mantissa.

The characteristic of the logarithm of any number to the base 10 can be written by inspection, as we shall now show.