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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; (ii) that involution and evolution are respectively

effected by the multiplication and division of the indices.

Example 1. If m=a", n=ak, then m xn=axab=ah+k

.(i), m;n=a"; ak= ah-k

(ii), m3=(ak)3=a?,

(iii). Vm=m* =(a;+ = ai Example 2. If 347= 102-5403295* and 461 = 1026637009, prove that

347 x 461=105-2040304. We have 347 x 461=102'5403295 x 102'6637009

102°5403295+2 6637009 =105-2040304

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

ndex 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) ma x n3. (ii) mt:n5, (iii) Wt x 15.

(iv) {'x 123/? (2) If 453=1026560982 and 650=1028129134, find the indices of the powers of 10 which are equal to (i) 453 × 650. (ii) (453) (iii) 6503 x 4532.

(iv) 1453. (v) 453 x 2650. (vi) 453 x (650). (vii) 1453 x 650.

(3) Express in powers of 2 the numbers, 8, 32, 3, t, 125, 128. (4) Express in powers of 3 the numbers, 9, 81, 3, , i, dr.

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=103-3750723, 4781 = 103-6795187, 2690=103-4297523,

Therefore 3759 x 4781 ; 2690=103-5750723 x 1030705187 ; 103-4297523 =103-5750723+3*0705187—3-4297523 = 103*8248387.

The list will give us that 103-8248387 =6680.9.
Therefore the answer correct to five significant figures is 6680-9.

Example 2. Simplify 36 x 210317601.
The list gives 2=10*3010300, 3=104771213 and 17601=1042455373.

Thus 36 x 210_) 17601 = (104771213,8 x (10-3010300) 10 = (1012455373);

= 102-8627278 x 103-0103000 = 101-4161791
= 102-8627278+3 0103000–1.4151791

=104-4578487.
And froin our list we find 101-4578487 = 28697, nearly.

EXAMPLES. LII.

Given that 2=10 3010300, 3= 10'4771213 and 7=10*8450980, find the indices of the powers of 10 equivalent to the quantities in the first 6 examples.

(1) 22, 34, 23, 2 x 3, 24, 72. (2) 14, 16, 18, 24, 27, 42. (3) 10, 5, 15, 25, 30, 35. (4) 36, 40, 48, 50, 200, 1000. (5) 310 x 710 :-220, 212 x 320 ;-711. (6) 21 x 18, 2:49 x 45 x 134 x 210.

(7) Find approximately the numerical value of V42 having given that 10-1623249 =1.4532 nearly.

(8) Find approximately the numerical value of N(42)* * \(42)3 having given that 103-38177=24086.

(9) Find the value (i) of 76x47xx9, (ii) of 1/2 x 3-4x711 having given that 10-8615067 =4.5868 and 10-0285094 = •93646.

(10) Find the value of (67.21)8 (49-62)} * (3-971)-'f having given that 67.21=101-8274339, 49•62=1016956568, 3.971 = 10-5988999 and 10.5971310=3:9549.

(11) Find the area of a square field whose side is 640·12 fcet; having given that 640*12=102-8062614 and that 1056125228 =40975.3.

(12) Find the edge of a solid cube which contains 42601 cubic inches; having given 42601=104-6294198 and 101-5431399 = 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 = 101-6806165 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 Loyarithms.

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 I be the logarithm of the number n to the base a, then

a=n.

193. The notation used is log, n=l.

Here, loga 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.'

Example 1. What is the logarithm of aš to the base a ?
That is, what is the index of the power of a which is af?
The index is ; therefore is the required logarithm,

log, al=.
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 ?

or

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

log, 32=5.

Example 3. Given that log10 2=-3010300, find log10 8 and log10 2000.

That log10 2=3010300, means that 10-3010300 =2, and

8=23=(10-3010300)3 = 109030900,

.: 8=100030900, or log10 8=:9030900. Again 2000=2 x 103 = 10-3010300 x 103

10-3010300+3=103-3010300 ; .: log10 2000=3:3010300.

EXAMPLES. LIII.

(1) Find the logarithms to the base a of a", a's', Ju, Ja, 1.

(2) Find the logarithms to the base 2 of 8, 64, 1, •125, *015625, 264.

(3) Find the logarithms to the base 3 of 9, 81, }, , 'i, 1 (4) Find the logarithms to base 4 of 8, V16, 45, V.015625. (5) Find the value of

log2 8, log2 ·5, log; 243, log('04), log10 1000, log10 ·001. (6) Find the value of

log.at, log, V6, log: 2, 109973, log100 10. If 2=10-3010300, 3=104771213 and 7=10*8450980, find the values of (7) log10 6, log10 42, log10 16. (8) logio 49, log1036, log10 63. (9) log10 200, log10 600, log10 70. (10) log105, log10 3:3, log10 50. (11) log1035, log10150, log10 -2. (12) log103.5, log10 7.29, log10 081. (13) Given log102, log103, log107, find the value (i) of 96 x 77 x 59, (ii) of 2 x 3-4x711

[.6615067 = log104.5868; – •0285094=log10 93646]. (14) Prove that (i) log {9/2 17+9)= }log 2+{log 7 - log 3,

(ii) log (W2 x 3-4x771) = 'u log 2-4 log 3+ log 7.

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