This last equation shows that the ratio of x to yz is the same for any two sets of corresponding values of x, y, and z. 623. If x varies directly as y and inversely as z, then x ∞ Let the student prove this. It is evident that if xx, we have an increase in x for an 2 increase in y, but a decrease in x for an increase in z. 21 EXERCISE 624. 1. If a varies jointly as y and z, and x1 = 63, y1 = 5, 9; find the constant of variation. = 2. By using the constant of variation found in example 1, Y2, if x2=72 and 22 = 18. find 225, Yı = 3. Given that x jointly as y and 2, and x1 = 12, z1 = 15, x=405, y2.6: find by proportion the value of z2. (See equation 3 of § 622.) 4. The total area T of a right circular cylinder varies jointly as R and R+ H, where R is the radius of the base and H is the altitude. When R = 7 inches and H 13 inches, T880 square inches; find T, when R = 5 inches and H = 10 inches. = 5. The weight of right circular cylinders of the same material varies jointly as the height and the square of the radius of the base. A steel cylinder weighing 22 pounds has a base with radius 1 inch and its altitude is 7 inches. Find the weight of another cylinder whose base has a radius of 2 inches and whose altitude is 14 inches. 6. The time required by a pendulum to make one vibration varies directly as the square root of the length. If a pendulum 100 centimeters long vibrates once in a second, find the time of one vibration of a pendulum 36 centimeters long. (Yale.) XXVIII. LOGARITHMS 625. The processes of multiplication, division, involution, and evolution can be greatly abridged by the use of the laws of exponents. A system of computation by means of tables is based upon these laws. 626. By means of a table of powers of 2 we can perform the operations of multiplication, division, involution, and evolution upon powers of 2. ORAL EXERCISE 627. 1. 32 × 128 = ? SOLUTION. From the table 32 = 25 and 128 = 27. = 212 ... 32 × 128 = 25 × 27 =25+7 = 4096. 219 = 524288. 220 1048576. 8. √65536 = ? 9. Divide 1048576 by 2048. 10. Divide 524288 by 512. 11. Divide 8192 by √1024. 628. Logarithm. The logarithm of a num ber is the exponent of the power to which a fixed number called the base must be raised to produce the number. Thus, in 2188192, 13 is the logarithm of 8192 to the base 2. This, in the notation of logarithms, is written log2 8192 = 13, and is read, the logarithm of 8192 to the base 2 is 13. Any expression of the form ac can be changed to logarithmic notation. Thus, abc and log, c = b according to the definition of logarithm, represent the same relation between a, b, and c. Read the following, and change each from the logarithmic nota Find the value of x in each of the following: 20. Solve for x, log10 100 = x. 19. log, 32 5. = 27. log16 8 = x. 630. Laws of Logarithms. The laws of logarithms for multiplication, division, involution, and evolution are exactly the same as the corresponding laws of exponents, as the student might anticipate, since logarithms are exponents. 1. Law of Multiplication. The logarithm of a product equals the sum of the logarithms of its factors. ... k* = a and k” = b. (Definition of logarithm.) .. log, ab = x+y, (Definition of logarithm.) k 2. Law of Division. The logarithm of a quotient equals the logarithm of the dividend minus the logarithm of the divisor. 3. Law of Powers. The logarithm of the power of a number equals the exponent of the power multiplied by the logarithm of the number. 4. Law of Roots. The logarithm of the root of a number equals the quotient of the logarithm of the number divided by the index of the root. 631. According to these laws we may make such transformations as the following: ab 1. log = + = с log a log blog c. (Why?) 2. log (ab)2 = 2(log ab) = 2 log a +2 log b. (Why?) 632. Using the laws of logarithms express examples 1 to 9 in terms of log a, log b, log c, and log x as in examples 1 and 2, § 631. |