47. Example 8. Divide .01265 by .55. Log. of .01265 = 2.1020905 Log. of.55 1.7403627 = Log. of .023 = 2.3617278 CHA P. IV. Of the APPLICATION of LOGARITHMS to CIRCULATING DECIMALS. 48. A S in fome Calculations, in which we propose to be very accurate, we may have Occafion to find the Logarithms of circulating Decimals; we shall explain the Method by which they may be found, as follows. The RULE. First turn the circulating Decimal into its equivalent vulgar Fraction, (as fhewn in Theorems 4, 5, 6, and 7, in Articles 13, 14, 15, 16, 17, 18, 19, 20, and 21, of the Effay on Circulating Decimals,) and then find the Logarithm of that vulgar Fraction, by Art. 21 of this ESSAY. 49. Example 1. What is the Logarithm of .4? By Art. 14. of the Effay on Circulating Decimals, •4 = $. Hence from Log. of 40.6020600 Subtract Log. of 90.9542425 Gives Log. of or of .4 1.6478175 50. Example 2. What is the Logarithm of 3.5 ? Solution. By Art. 16 of the Effay on Circulating Decimals, 3.5. From Log. of 350 = 2.5440680 51. Example 3. What is the Logarithm of .23? Solution. By Art. 18 of the Essay on Circulating 1 Decimals, .23 = 25. From Log. of 21 = 1.3222193 52. Example 4. What is the Log. of 26353.5? Solution. By Art. 20 of the Essay on Circulating Therefore, from Log. of 2609000 = 6.4164741 Subtract the Log. of 99 1.9956352 The Remainder is the Log. of 2609000 4.4208389 99 53. The Logarithms of Circulating Decimals being found by the above Method; Multiplication, and Divifion, are performed by them, after the fame Manner as is fhewn of finite Numbers in Chap. 2. and 3. of this ESSAY. CHAP CHAP. V. The GOLDEN RULE by LOGARITHMS. 54. FL ROM the Nature of the Golden Rule, and Logarithms, we have this Rule. Add the Logarithms of the fecond and third Terms together; from their Sum fubtract the Logarithm of the first Term; and the Remainder will be the Logarithm of the fourth, or required Term. 55. Suppofe in the Rule of Three Direct we have this Stating, As 4.1: 5: 65 a fourth Number.. 55. If from the Logarithm of 1 be fubtracted the Logarithm of any Number, the Remainder is called the Arithmetical Complement of that Logarithm. Hence, the Arithmetical Complement of the Logarithm of any Number, is equal to the Logarithm of the Quotient of I divided by that Number. 56. When it is required to divide one Number by another, it is in Effect the fame, if we divide by the Divifor, and multiply that Quotient by the given a I Dividend. (For = X a.) Which is done by m. 112 Logarithms, by adding the Logarithm of the Dividend to the Arithmetical Complement of the Logarithm of the Divifor. 57. Hence, the Operation of the Example in Art. 55, may ftand thus: From From Log. of 1 = 0.0000000 Sub. Log. of 4.1 0.6127839 Gives Arith, Comp. = 1.3872161 1.3872161 : 0.6989700 1.8129134 58. O raife any Number to a given Power by Logarithms, multiply the Logarithm of the Root by the Index of the given Power. Thus, for Inftance, if we are to fquare any Number, we muft multiply its Logarithm by 2; if to cube a Number, by 3, &c. and the Product (by Art. 4.) will be the Logarithm of the required Power. Note, If the Root be a Fraction, its Logarithm will be a Binomial; in fuch Cafe, the Carriage from the decimal Part in multiplying being affirmative, and the Product of the Charactcriftick negative, the Carriage must be subtracted from that negative Product, for the required negative Characteristick. 59. Example 1. What is the Square of 15? Log. of 15 1.1760913 2 Log. of 225 2.3521826 60. Example 2. Cube .55. Log. of .55 = 1.7403627 3 Log. of .166375 = 1.2210881 CHAP. |