r and the limit of r" or of arn is zero. Find the sum to infinity of + 1 + 1 + · 0.381=0.3818181 + ... =.3+.081+.00081+.0000081+. The terms .081 + .00081 + ... constitute a geometrical series whose common ratio is .01. Its sum is 9. a α 6. .8 .008 + .00008. 7. 10+180 + 1000+ · 8. 1. √2 2 + √2+1 3+2√2 Evaluate: (a) .36, (b) .63, (c) .90, (d) .80. 10. Evaluate: (a) .07846, (b) .006, (c) .0013, (d) .428571, (e) .1137, (ƒ) .0153846. 11. The sum to infinity of a G. P. is and the ratio is ; find the first term. 12. Find r if the sum to infinity of a G. P. is & and 13. The sum of the first 4 terms of a G. P. is 73 and the sum to infinity is 9. Find the first term. 14. The second term of a G. P. is 6 and its sum to infinity is 32. Find the first term. 15. Show that if 0 < x < 1 and if 8 = x + x2 + x3 + ... 16. Find the sum to infinity of (1+a)2 − (1 − a2) +(1 − a)2. a < 1. 17. Prove that the arithmetic mean of two numbers is greater than their geometric mean. CHAPTER XIX LOGARITHMS 201. If an, then x is the logarithm of n to the base a. This fact is expressed more shortly by the relation The logarithm of a number is the index of the power to which a number called the base must be raised to equal that number. 202. In practical work the base generally used is 10, and in stating logarithms it is as a rule omitted. Logarithms to base 10 are called common logarithms. Consequently the logarithm of a number between 1 and 10, i.e. of 1 figure = 0+ a fraction 10 and 100, i.e. of 2 figures 1+ a fraction = 100 and 1000, i.e. of 3 figures 2+ a fraction = Thus log 87.43 = 1 + a fraction, for 10 87.43 < 100. log 0.07382+ a fraction, for 0.01 0.0738 < 0.1. < The integral and decimal parts of a logarithm are called respectively the characteristic and mantissa of the logarithm. From the above illustration it is evident that the characteristic of the logarithm of a number greater than unity is one less than the number of figures to the left of the decimal point in the number, and that the characteristic of a number less than unity is negative, and numerically one more than the number of ciphers between the decimal point and the first significant figure. 203. Fundamental rules of logarithms. Let an, i.e. x = logan, ax+y= nm, i.e. x + y = log。 nm = logan + log.m I III arx = n', i.e. rx = = loga n2 = r loga n. In words: The logarithm of a product is the sum of the logarithms of the factors. I The logarithm of a quotient is the logarithm of the dividend minus the logarithm of the divisor. II The logarithm of a power is the logarithm of the number multiplied by the exponent of the power. III log 7 3/5 = log 7. 53 = log 7 + † log 5, log mnr = = 1(log m + log n + log r — log 8). The mantissæ of the common logarithms of numbers expressed by the same significant figures are the same. log 25743.41061, log 257.4 = log (2574 × 10−1) = 3.41061 − 1 = 2.41061, log 25.74 = log (2574 × 10-2) = 3.41061 - 21.41061, log 2.574 = log (2574 × 10-3) = 3.41061 - 30.41061, log 0.2574 = log (2574 × 10-4)=3.41061-4=0.41061-1, log 0.02574= log (2574 × 10-5)=3.41061-5=0.41061-2. In practice, logarithms are written so that the mantissæ are positive. Example 1. What is the common logarithm of (√.001)5? (√.001)5 = (√10−3)5 (10)5 — 10-7.5. = = Ans. —7.5. - 7.5 is written 8.5, for- 7.5 = 8 — 7.5 – 8 = 0.5 – 8. Example 2. Given log 2.30103, log 3 =.47712, find log 24. or log 24 = log (23 × 3) = 3 log 2 + log 3 = 3(.30103) +.47712 = 1.38021, 2428 × 3 = (10-30103)8 × 10-47712 — 101.38021 .. log 241.38021. Example 3. Given log 17 = 1.23045, log 992=2.99651, find log. log log 17 - log 992 1.230452.99651 992 |