then

CHAPTER XIV.

ON LOGARITHMS.

189. IN Algebra it is explained

(i) that the multiplication of different powers of the same quantity is effected by adding the indices of those powers;

(ii) that division is effected by subtracting the indices;

(iii) that involution and evolution are respectively effected by the multiplication and division of

the indices.

Example 1. If m=a", n=a*,

Example 2. If 347=1025403295* and 461=1026637009, prove that

We have

347 x 461=105-2040304

347 x 461=1025403295 x 102-6637009

=1025403295+2*6637009
=105-2010304

Q. E. D.

* The number 347 lies between 100 and 1000, i. e. between 102 and 103. Hence, if there is a power of 10 which is equal to 347, its index must be greater than 2 and less than 3, i.e. equal to 2+ a fraction.

EXAMPLES. LI.

(1) If m=a", n=a*, express in terms of a, h and k,

(i) m2 × n3. (ii) m1÷n5. (iii) m2 x 25.

(iv) {√m3 × n3}2.

(2) If 453=102·6500982 and 650=102‍8129154, find the indices of

the powers of 10 which are equal to

(i) 453 × 650. (ii) (453)4. (iii) 6503 x 4532.

(v) √453x650.

(vi) 453 × (650)3.

(iv) 453.

(3) Express in powers of 2 the numbers, 8, 32, 1, 1,

125, 128. 1, 27, 1, §1.

(4) Express in powers of 3 the numbers, 9, 81, 3, 27,

190. Suppose that some convenient number (such as 10) having been chosen, we are given a list of the indices of the powers of that number, which are equivalent to every whole number from 1 up to 100000

Such a list could be used to shorten Arithmetical calculations.

Example 1. Multiply 3759 by 4781 and divide the result by

2690

Looking in our list we should find 3759-1035750723, 4781 =103 6795187, 2690=1034297523

Therefore 3759 × 4781 ÷ 2690=1035750723 × 1036795187 ÷ 1034297523 =1035750723+3*6795187-3*4297523=1038248387

The list will give us that 1038248387=6680·9.

Therefore the answer correct to five significant figures is 6680.9.

Example 2. Simplify 36 x 210/17601.

The list gives 2=103010300, 3=104771213 and 17601=104-2455373

Thus 36 x 21017601=(10)4771213)6 × (103010300)10÷(104·2455373) =102-8627278 × 1030103000101-4151791 =102·8627278+3*0103000-1-4151791

=104 4578487

And from our list we find 1044578487=28697, nearly.

Given that 2=103010300, 3=104771213 and 7=108450980, find the indices of the powers of 10 equivalent to the quantities in the

(7) Find approximately the numerical value of 42 having given that 101623249=1·4532 nearly.

(8) Find approximately the numerical value of √(42) × √ (42)3 having given that 103-38177-2408'6.

N

(9) Find the value (i) of 3/6 × 7 × 5′9, (ii) of 12 × 3-1 × 71 having given that 10 6615067=4.5868 and 10-0285094;

93646.

(10) Find the value of (67·21) × (49·62)* × (3·971)-* having given that 67.21=101‍8274339, 49·62=101·6956568, 3·971=10·5988999 and 10 5971310-3.9549.

(11) Find the area of a square field whose side is 640 12 feet; having given that 640·12=102-8062614 and that 105 6125228=40975.3.

(12) Find the edge of a solid cube which contains 42601 cubic inches; having given 42601=104-6294198 and 1015431399-34.925.

(13) Find the edge of a solid cube which contains 34.701 cubic inches; having given that 34-701=1015403420, and 10'5134473-3.2617.

(14) Find the volume of the cube the length of one of whose edges is 47.931 yards; having given that 47·931=1016806165 and that 105 0418195=110115.

191. The powers of any other number than 10 might be used in the manner explained above, but 10 is the most convenient number, as will presently appear.

192. This method, in which the indices of the powers of a certain fixed number (such as 10) are made use of, is called the Method of Logarithms.

Indices thus used are called logarithms.

The fixed number whose powers are used is called the base. Hence we have the following definition :

DEF. The logarithm of a number to a given base, is the index of that power of the base, which is equal to the given number.

Thus, if be the logarithm of the number n to the base a, then a'=n.

193. The notation used is

logan = l.

Here, log, n is an abbreviation for the words 'the logarithm of the number n to the base a.' And this means, as we have explained above, the index of that power of a which is equal to the number n.'

or

Example 1. What is the logarithm of a to the base a ?

That is, what is the index of the power of a which is a ? The index is ; therefore is the required logarithm,

log. a=§.

Example 2. What is the logarithm of 32 to the base 2 ?

That is, what is the index of the power of 2 which is equal to 32 ?

Now 32=25. . the required index is 5; or

log2 32=5.

Example 3. Given that log10 2=3010300, find log108 and log10 2000.

(1)

Find the logarithms to the base a of a3, a's, Vu, Va2,

(2) Find the logarithms to the base of 8, 64, 1, 125, 015625, 64.

(3) Find the logarithms to the base 3 of 9, 81, 3, 27, 1, §. (4) Find the logarithms to base 4 of 8, 16, 5, 015625. (5) Find the value of

log, 8, log, 5, log, 243, log (04), log10 1000, log10 001. (6) Find the value of

log.a, log, 2, log. 2, log27 3, log100 10.

If 2=10 3010300, 3=10'4771213 and 7-10 8450980, find the values of

(8) log10 49, log10 36, log10 63. (10) log10 5, log10 3.3, log10 50. (12) log10 3.5, log10 7-29, log10 081.

(7) log10 6, log10 42, log10 16. (9) log10 200, log10 600, log10 70. (11) log10 35, log10 150, log10 2. (13) Given log10 2, log10 3, log10 7, find the value (i) of 3/6 × 7×59, (ii) of 2×3-x714

[6615067=log10 4.5868;-0285094=log10 '93646].

(14) Prove that (i) log (3/2 × √7÷5/9}=log2+log 7 — 2 log 3, (ii) log (W2 × 3-4 × 771) = log 2 - log 3+ log 7.