(11) If a be positive and less than unity, and if a be the least value of sin-1a, then (12) Prove that tan A and sin 24 have always the same sign. Solve the six following equations. (13) cos A+ cos 34 + cos 54=0. (14) sin 50+ sin 30+ sin 0=3-4 sin2 0. (15) 2 sin2 34+ sin2 64 = 2. (16) a (cos 2x − 1)+2b (cos x+1)=0. (17) sin (m+n) 6+sin 2m 0+sin (m − n) 0=0. (18) sin {xx (x+y)}+sin(xy (x+y)} = 0; } sin x2+sin у2=0. (19) Trace the changes in the sign and magnitude of the following expressions, as changes from 0 to π. have exactly the same series of solutions. (21) Explain why exactly the same series of angles are given 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 2010304 347 x 4611025403295 × 1026637009 =1025403295+2*6637009 =1052010304 Q. E. D. * The number 347 lies between 100 and 1000, i. e; between 10% 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 x n3. (ii) m1÷n5. (iii) m2 x no. (iv) {m3 xn32. (2) If 453-1026560982 and 650=1028129134, find the indices of the powers of 10 which are equal to (i) 453 × 650. (ii) (453)1. (iii) 6503 × 4532. (vi) 453 × (650)3. (iv) 453. (vii) 453 × 650. (v) √453 × √650. (3) Express in powers of 2 the numbers, 8, 32, 1⁄2, 1, 125, 128. (4) Express in powers of 3 the numbers, 9, 81,,, ·1, §. 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 =1036795187, 2690=1034297523 Therefore 3759 x 4781 ÷ 2690=1035750723 x 1036795187÷103:4297523 =1035750723+36795187-34297523=103-8248387. 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 × 210÷17601=(10)4771213)6 × (103010300)10÷(104·2455373) § =1028627278 × 1030103000÷101-4151791 And from our list we find 104457818728697, nearly. EXAMPLES. LII. Given that 2=103010300, 3=104771213 and 7=10'8450980, find the indices of the powers of 10 equivalent to the quantities in the (6) 21 x 18, 49 x 45 x 31 x 210 (7) Find approximately the numerical value of 42 having given that 10'1623249=1.4532 nearly. (8) Find approximately the numerical value of (42) × √(42)3 having given that 10338177-2408'6. (9) Find the side of a square field which contains 73401 square yards; having given that 73401=1048657020 and that 1024328510270.926. (10) Find the side of a square plot of ground which contains 54331 square feet; having given that 54331=1047350477 and that 102 3675238233.09. (11) Find the area of a square field whose side is 640 12 feet; having given that 640·12=1028062614 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 1050418195110115. 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. then Thus, if I be the logarithm of the number n to the base a, 193. The notation used is log, n = 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, a1=ş. 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 log, 32=5. |