CHAPTER IX. LOGARITHMS. 74 75 THE calculations necessary to the finding of the parts of triangles, by the use of these formulæ, would be long and inconvenient, were it not for the use of logarithmic tables. By the use of these the operations of mathematics are much simplified. Multiplication of two numbers is replaced by the addition of their logarithms; Division by the subtraction of their logarithms; while the tedious operation of finding roots and powers is effected by the division and multiplication of the logarithms by the indices. A logarithm is an index. If a*=m, x is called the logarithm of m to the base a, and it will be seen that the operations spoken of are just those which take place with the indices of numbers. The best definition of a logarithm that can be given is alogax = x, when logx is read "the logarithm of x to the base a." Thus 10-201600=2. Here 301030 is the logarithm of 2 to the base 10, the index of the power to which 10 is raised. Using this definition we have Multiplying, Dividing, x = a logar y = ɑlogy. alog = xy=alog++logy. ..log.xy=log.x+logy. (1) 76 77 Raising to the power m, alog = x” = amlogTM ̧ ..log.x = m logx. (3) 1. The logarithm of a product is equal to the sum of the logarithms of the factors, as 2. The logarithm of a quotient is equal to the difference of the logarithms of the factors, as 3. The logarithm of a power is equal to the product of the index of the power, and the logarithm of the number, as Tables are computed which give the logarithms of numbers, and the logarithms of the sine and cosine of angles less than 45o. All the preceding formulæ may be written in the form adapted for the use of tables, if they are in the form of products. 78 The logarithm of a sum is not the sum of logarithms, so that for this method of computation those which involve sums only are useless, e.g. A 1 A = (s — b) (s — c) § 67. 2 =(log (sb) + log (sc) - log blog c}. 2 2 log sin 1 A log tan (B-C) = log (b − c) — log (b + c) + log cot 2* In referring to a table of logarithms, numbers will be found in this form: 825 916454 6507 6559 and so on for 3, 4, &c. This means that A. T. log 8250 = 3.916454, log 8251 = 3.916507, log 82523916559. པ་ 5 79 The numbers given in the table are decimals and are the logarithms of 8'250, 8°251, and so on, for numbers which lie between 0 and 10 must have logarithms between 0 and 1, that is, pure decimals. A number is placed before these decimals which depends on the value of the place of the highest digit. This is called the characteristic, since it denotes the integral part of the power of 10. If this place be 10", for instance millions, the characteristic is n, in that case 6, log 8252000=6.3916559. If in a decimal the highest place be 10, the characteristic is n; tens of thousandths giving 4. The theory of logarithms simply is that which has been given here, that a logarithm follows the rules of indices; but it will be understood that the use of logarithms can only be mastered by practice. Those who wish to follow out the practical part of the subject will now take up books on Mensuration. Prof. Elliott's books are good instances of the kind. Those who wish to attend to the theoretical questions of Trigonometry will find that what has been said here will be a stepping stone to Todhunter's larger Trigonometry. MISCELLANEOUS EXAMPLES. I. 1. SHEW that neither the sine nor the cosine of an angle can be greater than unity. 2. Prove that cot2 A + cosec2 A = (sin2 A + 2 cos2 A) cosec2 A. 3. Find sin 25o and cos 15° from the ratios of 4. At what height will a bullet hit a house, when it is fired from the ground 20 yards off, at an elevation of 15° ? 5. What are the measures of the angle of a quindecagon ? 6. If one side of a triangle be 15 ft. and the adjacent angles be 60° and 75°; find the other angle and the sides. 7. What is the angular elevation of the sun when a vertical stick casts a shadow as long as itself? |