MATHEMATICS. ALGEBRA. 



[l-SE OF LOOARITIIM-i. 



the approximate value U the only one we can get, and 

 tin tenet it (At maiu 6y uAt>A IM yet U. For instance, 

 we hare called the base of the Napierian logarithm . 

 But what is I It U iuch a number Uiat 



1. (._!)_}(,_ 1)1 + Ka_)S_ Ao . 



This U an equation we have no method no direct 

 method of solving. We have seen, however, that e is 

 expressed by the series 



From this, a* we have already aeon, we can find that 

 37183818, <tc. The student may think an approxi- 

 mation a very unsatisfactory result ; but he must re- 

 member two thing*: (1). That greater part of the 

 quantities we have to deal with cannot be expressed in 

 whole numbers, or in vulgar fractions e.g., so common 

 and elementary an expression as J 2 cannot be expressed 

 as a vulgar fraction ; and (2), that in practice no mea- 

 surement U or curate, but is known to lie within certain 

 limits. For instance, if a tailor measure a piece of cloth, 

 he calls it a yard, though it may happen to be a quarter 

 of an inch more or less. In like manner the most re- 

 fined scientific measurements (the length of the second's 

 pendulum, of an arc of the meridian, etc.), are generally 

 the meant of several results, and are accurate to within 

 certain very small limits. Now, in approximating to a 

 result by means of a series, we can always get to within 

 any given limits that may be assigned. And thus ap- 

 proximations by means of series are as accurate as any, 

 the most refined, measurements can be. 



In practice, if we know a number to be true for the 

 first six or seven places of decimals, it is generally known 

 with sufficient or even more than sufficient accuracy. 

 Thus, if we are certain that x. lies between 316764 aud 

 3 16755, we may call it 3 10754 ; although if we calculated 

 to a greater nicety we might obtain z = 3 '1676438295, 

 for the error we commit is < tooioooi *?> an error less 

 than \ of an inch in one mile. 



It is to be observed, that in calculating the value of a 

 series, we must calculate each term to one or two more 

 places of decimals than the result we wish to obtain, so 

 as to be quite sure that we carry the right number to 

 the seventh place. Thus, in finding the value of e 

 (Art. 11), we calculated each term to nine places of 

 decimals, to ensure that our result should be true to 

 seven places. 



It U also to be observed, that in cutting off the eight 

 and subsequent decimal places, if the eighth place be 

 5, 6, 7, 8, or 9, we add 1 to the seventh place ; but if 

 4, 3, 2, 1, or 0, we simply omit it. Thus we reckon 



2-50710345827 = 2-5971G35 

 But 2-59716343254 = 2-5971034. 



For it is plain that 2-59716345827 is nearer to 

 2-5971635 than to 2-5971634 : whereas, as in the second 

 instance, the contrary is the case. 



We now proceed to consider the subject of logarithms 

 in detail. 



14. To Explain the Principle on which Logarithms may 

 be rued to Facilitate Calculations. 



From the definition of a logarithm already given, it 

 follows that if M = o* then x is the logarithm of M to 

 the base a ; and if N = o then y is the logarithm of N 

 to the base o. Now observe M X N = o* + ", whence it 

 is plain that the multiplication of one number by another 

 corresponds to the addition of their logarithms. In like 

 manner, M -=- N = a*-, or the division of one number 

 by another corresponds to the subtraction of the loga- 

 rithm of the dividend from that of the divisor. Again, 

 M" a"*, or the raising of a number to a given power 

 corresponds to the multiplication of the logarithm by 



i * 



that power. In like manner MS o, or the extract i. .n 

 of the root of a given number corresponds to the divi- 

 sion of the logarithm by that root. So that if we knew 

 the logarithm which corresponds to any number what- 

 ever, and wished to find the product of two numbers, we 



should moruly have to write down the logarithms of the 

 iiiiiiil.i-r.-i, a<U tli. -in, and then the number whose loga- 

 rithm is that sum will bo the product of the two given 

 numbers ; and similarly for the other rules. 



Now, tables have been calculated which give us the 

 logarithm corresponding to any number between 1 and 

 10,000,000. Hence, by using these tables properly. 

 multiplication is performed by means of addition ; and 

 in like manner division by means of subtraction, in- 

 volution by multiplication, and evolution by division. 



In the following pages we shall first explain the 

 method by which these tables are calculated, and then 

 proceed to show how they are practically employed.* 



16. The following results follow manifestly from what 

 has been slid. 



(1). That if P - Q, then log..P = log..Q. 

 (2). Since o 1 a. .-. log. a o = 1. 

 (3). Since a" - 1. .-. log..l = 0. 

 (4). If M = a* , N = o , then x = log..M, y - log..N. 

 Now, JIN = a* X a* a'+r .'. x + y log..MN 



. . log..M + log..N = log..MN. 



- 



(5). Similarly, a*H-a v a*-*, . 

 i M 





log. 31 log.. N - log..". 



(6). Again, if a- M. . . log..M = x. 

 Now, M" = a* . . log. M = mx. 

 . . m log..M = log..M,". 



(7). Similarly, M" a~ 



- log..M = log. .Ms 

 m 



(8) . From (4) it is manifest that 

 log..M + log..N + log..P + log .MNP 



16. To show tlult log J>, + log.z = log..j;. 



Now, suppose x = a", and x = b" 

 . . log..x = m, and log.*! = n. 



But since a" = b" wa must have a = 6T 

 . . Io '. a = log. "6S 



n. W..b = m 

 or log. .6 log.z = log..x. 



From this it follows, that if we know the logarithm of 

 a given number to a given base, we can find its logarithm 

 bo another base, by dividing the first logarithm by the 

 logarithm of the now base ; for instance, suppose our 

 tables give the logarithms of numbers to the base 10, 

 and suppose we wished to find the logaritlun of a given 

 number (N) to the base 19, we have 



log., 19 log. , 9 N = log, , N. 



Then log. , p !9 and log. , N T are given by the tables, and 

 therefore we know log. 19 N by division. 



17. The practical advantage of Calculating Logarithms 

 to tlui Base 10. 



The tables of Logarithms tommonly printed, are 

 logarithms to the base of 10. In all future articles, 

 whenever we write log.sc, we mean logarithm to the 

 aase 10. 



We might calculate tables to the base e ; and the 

 calculation is obviously rendered much easier when this 

 uaso is employed by the circumstance that 



whereas, if we use the base 10. we have 



log. .10. log. x - "5 ^j- + 3 ~ &c - 



An extended table of logarithm! will be found at uie end of thl 

 chapter. 





