LESSONS IN LOGARITHMS. 



MB 



LESSONS IN LOGARITHMS. III. 



COMMON SYSTEM OF LOGARITHMS. 



34. To find the Logarithm of any Prime Number. Bole 1. 

 Divide the given prime number by the natural number nearest 

 to it in the skeleton tables, but leas ; divide the quotient by 

 the natural number nearest to it, but less ; divide thia quotient 

 by the natural number nearest to it, but less ; and BO on, till 

 the last quotient coincide with some natural number in the 

 tables; then, the lost quotient with all the divisors are the 

 tabular factors of which the prime number is composed. Con- 

 sequently, if the logarithms of all those factors, given in the 

 tables, be added together, their sum will bo the logarithm of 

 the given prime number. On this principle the following table, 

 exhibiting the method of calculating the logarithm of the prime 

 number 2, is constructed : 



FIRST CALCULATION OF THE LOGARITHM OF 2. 



Ptrtdands 



V12468 



ram 



1-00961 

 1-00057 



Pivinors. 

 177S-.W 

 1-07461 

 1-03663 

 l'0090i 

 1-00056 

 1-00001 



Quotients. 

 112468 

 1-04660 

 1-00961 

 1-00057 

 1-00001 

 1-00000 



Logs, of Diviners. 

 250000 

 031250 

 015625 

 003906 

 000244 

 000004 



Logarithm of 2 = Sum -301029 



35. To find the Logarithm of any Prime Number. Rule 2. 

 Look for the tabular number nearest to the given prime number, 

 but greater ; divide the former by the latter ; divide the quotient 

 by the tabular number nearest it, but less ; and so on, as before, 

 till the last quotient coincide with some tabular number ; then, 

 the last quotient with all the divisors but the first are the 

 tabular factors of the first quotient. Consequently, if the sum 

 of the logarithms of these factors, which is the logarithm of the 

 first quotient, be subtracted from the logarithm of the first 

 dividend, the remainder will be the logarithm of the given 

 prime number. On this principle, the following table, exhibit- 

 ing another method of calculating the logarithm of 2, is con- 

 structed : 



SECOND CALCULATION OF THE LOGARITHM OF 2. 



Logarithm of 2 = Remainder "301030 



The latter logarithm of 2 is more correct than the former, 

 owing to the difference in the mode of calculation. The loga- 

 rithm of 2, calculated to ten places of decimals, is '3010299957. 



36. As the prime number 5 is the quotient of 10 divided by 

 2, its logarithm is found on the principle that if the logarithm 

 of the dividend be subtracted from the logarithm of the divisor, 

 the remainder is the logarithm of the quotient (see Art. 20). 

 Hence the reason of the following calculation is made evident : 



Logarithm of 10 = 1 -000000 

 2 = -301030 



5 = -698970 



37. By the application of either of the preceding methods, or 

 by a judicious combination of both, the logarithms of all the 

 prime numbers to any extent may be found. The following 

 table exhibits the logarithms of some prime numbers, which 

 may be calculated in the manner proposed : 



LOGARITHMS OF PRIME NUMBERS. 



Natural Numbers. 

 3 . 

 7 . 

 11 



17 



Logarithms. 



2-004321 



3-005609 



etc. 



38. The logarithms of the powers of a prime number are 

 found by multiplying its logarithm by the indices of those 



power* (ee Art 22). 

 are oonntructed : 



On thu principle, the following t*b!* 



LOGARITHMS OF TUB POWERS OF 2. 



Log. 4 - a x -301030 - -602060. 

 8 - 3 x -301030 - -903000. 

 16 = 4 x -301030 = 1-204120. 



Log 32 - 5 x -301030 . 

 04 - * 



etC. ''I 



LOGARITHMS OF THE POWERS OF 3. 



Log. 243 - 5 x -477121 

 = 6 x 477121 

 etc. etc. 



: - '.- 

 . - I 

 etc. 



etc. 



f -1 I v 

 19). Ou 



Log. 9 o 2 x -477121 = -954243. 

 87 - 3 x 477121 = 1-431304. 

 81 - 4 x -477121 = 1-908485. 



39. The logarithms of the composite numbers are 

 the addition of the logarithms of the factors (sec Art. 

 this principle, the following table is constructed : 



LOGARITHMS OF COMPOSITE NUMBERS. 

 Log. 6 = log. 2 + log. 3 = -778151. 

 12 = 2 + 6 = 1 079181. 

 18 = 3 + 6 = 1-255273, etc. 

 Log. 14 = log. 2 + log. 7 = 1 146128. 

 21 = 3 + 7 = 1-322219. 

 28 = 4 + 7 = 1-447153, etc. 

 Log. 15 = log. 3 + log. 5 = 1 176091. 

 20 = ., 2 + 10 = 1-301030. 

 25 = 5 + 5 = 1 397940, etc. 

 Log. 105 = log. 3 + log. 5 + log. 7 = 2-021189. 

 385 = 5 + 7 + 11 = 2 585461. 

 1001 = 7 + 11 + 13 = 3000434, etc. 



40. The integer prefixed to the decimal part of a logarithm ur 

 called its index or characteristic. Thus, in the preceding table, 

 the logarithm of 20 is 1*301030, of which 1 is the index or cha- 

 racteristic, and -301030 is the decimal part or mai.t 



41. From the skeleton tables and the preceding articles, it in 

 evident (1.) that the index of the logarithm of every number 

 between and 10 is ; the index of the logarithm of every 

 number between 10 and 100 is 1 ; the index of the logarithm of 

 every number between 100 and 1000 is 2 ; and so on. Hence, 

 generally, the index of the logarithm of every integer is a number 

 less by unity than the number of figures which it contains. The 

 index of the logarithm of a mixed number, being determined 

 solely by its number of figures, is, of course, not affected by 

 the decimal. 



42. (2.) The index of the logarithm of every decimal of 

 which the highest place is tenths is - 1 ; the index of the 

 logarithm of every decimal of which the highest place is hun- 

 dredths is 2 ; thousandths, -3; and so on. Hence, generally. 

 the index of the logarithm of every decimal is a number denoting 

 its highest place, with a negative sign attached to it. The use of 

 this sign, which is usually written above the index, is to indicate 

 that when the logarithm of a decimal is added, its index is to 

 be subtracted, and when the logarithm of a decimal is sub- 

 tracted, its index is to be added. 



43. In tables of logarithms, only the decimal parts or man- 

 tissffi of the logarithms of the natural numbers are printed ; 

 hence, the preceding rules for supplying their indices are indis- 

 pensably necessary for the purpose of calculation. To facilitate 

 this process, however, the following table is added : 



TABLE OF INDICES OF LOGARITHMS. 



Part I. 



For Integers. MID 



Tens of Millions . . .7 

 Hundreds of Millions . . 8 

 Thousands of Millions . . 9 

 Tens of Thousands of Millions 10 

 Hundreds of Thousands of Mil- 

 lions . . . .11 

 etc. etc. 



For Decimals. 



Hundredths of Millionths . 8 

 Thousandths of Millionth* . 9 

 Tenths of Thousandths of Mil- 

 lionths . . . .10 

 dredths of Thousandths 

 of Millionths. . . 11 

 etc. etc. 



