Conse will have the number 2.718281828, for the sum of the entire infinite series, true as far as nine places of decimals. quently A is such a number that 1 a=2718281828; or that (2.718281828)^=a. Without at present seeking to compute A, let us represent the known value of aa, namely, 2-718281828, by e; then by (6) x1 e=1+x+ + + + &c...... (8). 2 2.3 2.3.4 We have thus got the exponential theorem, in a form perfectly general as respects the exponent x, but restricted as respects the base a; the particular value of the base, represented for distinction sake by e, is limited to the single value, 2.718281828. To give to the theorem the desired extension as respects the base a, requires the aid of a principle now to be established. THE THEORY OF LOGARITHMS. 116. If every positive number be considered as generated from some power, whole or fractional, positive or negative, of any assumed number fixed upon at pleasure, then, instead of using the ordinary notation, every positive number might be written in the form of a power-a power of the assumed number or base originally chosen. And a new problem would be presented to us, namely: the base a being given, and any number n being proposed, to find the value of x, such that the equation a*=n may be true. The general solution of this problem for all values of n, a being invariable when once fixed upon, would enable us to effect the change of notation just adverted to; every positive number might then be represented by the fixed number a, with a certain known exponent over it. It is this exponent which is called the logarithm of the number; or, rather, the logarithm of the number according to the proposed base a; for different bases or values of a of course require different exponents or logarithms to justify the equation an. Napier, the inventor of logarithms, chose the number e, determined above, for the base, namely, the number 2.71828, &c.; but this number was afterwards changed by Briggs for the more convenient number 10; and the logarithms or exponents are now computed and arranged in tables always in reference to the base 10; that is, M the exponent or logarithm x, for any number n, is always that furnished by the solution of the equation 10*=n; so that the statement "6 x is the logarithm of n," means, in reference to modern tables of logarithms, that x is the proper exponent to be placed over 10, to give n; and the expression is more briefly written, x=log n. As just remarked, Napier computed his exponents or logarithms in reference to the base e; and although his table has become superseded by that of Briggs, for general purposes it is nevertheless not without its value: it is of service to us, for instance, in our endeavours to generalize the exponential theorem, considered in the preceding article, as we shall see when a few obvious propositions in the theory of logarithms are stated. 117. The sum of the logarithms of any two numbers is equal to the logarithm of their product. Let an, and a*'=n' .. a*+*'nn'; therefore, if a be the base of the system of logarithms, x=logn, x'=logn', and x+x'=log nn'; that is, log n + log n'=log nn'. In like manner, log nn'+log n"= log nn'n' = log n+log n'+log n"; and so on for any series of numbers. Hence the sum of the logarithms of any numbers, however many, is the logarithm of their product. If, therefore, a table of logarithms be formed of sufficient extent, the multiplication of numbers may be avoided: we have only to take the logarithms of the factors out of the table, to add these logarithms together, and then to seek in the table for the number answering to their sum. 118. If n, n', n", &c. are all equal, and are m in number, then the sum of their logarithms is m logn, and the log of their product is log n"; so that m times the logarithm of a number is the logarithm of its mth power. 119. The difference of the logarithms of any two numbers is equal to the logarithm of their quotient. n n x-x' =a* = .. x-x'=log. that is, log n-log n'= η n n' log Hence, by help of a table of logarithms, division of n' numbers may be dispensed with: we have only to subtract the log of the divisor from the log of the dividend to get the log of the quotient: the number answering to this latter logarithm in the table is the quotient itself. 120. The property in (118) may be generalized. It is true that log n"=m log n, whether m be whole or fractional, positive or negative. For let k be the logarithm of n; that is, put a for n, then log n log ak―mkm lbg n. Therefore, to find the logarithm of any power m of a number, we have only to multiply the logarithm of that number by the exponent m, whatever m may be. Logarithms to the Napierian base e, are usually distinguished from those to the common base 10, by prefixing the abbreviation Nap. to the abbreviation log; thus Nap. log a means the Napierian logarithm of a. And since, as shown above, 1 A 1 a =e.. Nap. log a=Nap. log e=1.. A=Nap. log a; and thus A is determined by aid of a table of Napierian logarithms. Hence the exponential theorem, in its most general form, is aa=1+ Nap. log a x+ (Nap. log a)2 (Nap. log a)3 x3+ &c. 1 2 2.3 121. The Logarithmic Theorem is a theorem for the development of log (1+x), according to the ascending powers of x; if this be possible, the form of the development must be log (1+x)=A1x+А ̧‹æ2+Аzï3+a ̧æ1+ &c..............(1); because, if x be put =0, the first member becomes log 1, which is 0, since ao=1; so that the development cannot contain any term independent of x. By changing x into x+, we have log (1+x+≈)=A,(x+≈)+A2(x+≈)2 + A3(x+≈)3+ A1(x+2)1+ &c.............. (2). 1 But 1+x+z=(1+x)(1+ 2).. log (1+x+2)= 1+x log (1+2)+log (1+1+2 2). Consequently, by equation (1), log (1+æ+x)=A‚ ̧æ+A ̧æ3+A2x2+ &c. + A11 + x^ 2+ therefore the second member of this equation must be the same as the second member of (2), and the complete coefficient of z in (2), as found by applying the binomial theorem to the several terms, must be the same as the complete coefficient of z in (3). (See page 237.) The complete coefficient of z in (2) is obviously the expression A,+2A+3A3+4A,3+ &c.; and the complete coefficient of ≈ in (3) is simply A, 1 1 + x A-A1x+A ̧x2-A,+ &c. because the terms beyond this in (3) contain only the higher powers of z; consequently, these two expressions being identical, we have, (page 237), AA, 2A,—A, .:. A2=—A1, 3A, A, .. A=A1, 4A ̧=—A, .: A1——‡Ã„, &c.; Put .. log (1+x)=A‚(x—†x2+}x3—†x1+ &c.)..............(4). 1 A1 = (a—1) — 1 (a—1)2 + ¦ (a—1)3 — (a—1)'+&c. Hence A, in (4) becomes known as soon as the base a of the system of logarithms is fixed upon: we see that it is upon this base that the value of A, entirely depends. There is an intimate relation between this A, and the A which represents the coefficient of x in the development of aa:—the one is the reciprocal of the other; that is, -A; in other words, (a—1)— Αν 1 } (a—1)2+} (a—1)3—4 (a—1)1 + &c. is the Nap. log a, as may be thus shown. It follows from (4), that 1+x=aa1(x−}x2+}x3—‡x1+ &c.), because, taking the logarithm of each side, (4) is the result. The right hand member of this equation, when developed by the exponential theorem (p. 243), changes the equation into 1+x=1+A, Nap. log a.(x-x2+&c.) 2 (A, Nap. log a)2 Now, as the two members of this are identical, their equality subsisting whatever be the value of x, it follows (p. 237) that all the terms involving powers of a higher than the first must vanish; for no such higher powers exist in the equivalent first member. Consequently the equation is in reality nothing more = Nap. log a; than 1+x=1+ A, Nap. log a.x .. 1 Α, .. Nap. log a (a−1)-(a−1)2+}(a—1)3— (a-1)'+ &c. If a be equal to e, the Napierian base, then 1=(e-1)-(e-1)2+}(e—1)3—(e-1)'+ &c. which is therefore the value of A, in (4), when the Napierian logarithms alone are considered; so that the general expression for log (1+x), in this system, is simpler than it would be for any other system: a reason which prevailed with Napier in his choice of a base. CONSTRUCTION OF A TABLE OF LOGARITHMS. 122. Resuming the general equation (4), which is not restricted to any particular system, that is, to any particular value of the base a, we have, by putting for x any number, n,— log(1+n)= A1(n—}n2+}n3—\n1+ &c.); ·.log(1—n)= A‚(—n—\n2—}n3—\n1— &c.); .. ..log(1+n)-log (1-n)=2A,(n+}n3+1n5+ &c.)= To render this series more rapidly computable, we must change it into another of which the terms more rapidly decrease; that is, we must render it more convergent, as it is called. We 1 + n for n, which will change 1. -n p p 2A, 1 + &c.) + log p. 2A. (2p+1 1 2p+1'3(2p+1)31 5(2p+1)5 1 1 (2p+ 1 + 3(2p+1) 3 From which it appears that if p be any of the numbers 1, 2, 3, &c. the terms of the series will rapidly diminish in value, and that when log p for any one of these values of p is known, the value of log (p+1) for the next number may be computed to any proposed extent of decimals. Now log 1 is known, whatever be the base; it is 0 (see Art. 121); hence, supposing at first that A,1, or that the logarithms are Napierian, we have, by putting p1, 2, 3, &c. in succession, the following |