98. If in (48) x = log. b in any system, x'= L. b,* in the naperian system, and M'= modulus of naperian system =1, we have log. b : L.b :: M:1, whence log.b=MXL.b. (49) That is, the logarithm of a number in any system is equal to the modulus of that system multiplied by the naperian logarithm of the number. 99. If b=a= base of the system whose modulus is M, then log. b=l, and (49) becomes 1=Mx L. a, 1 whence M= (50) L. a That is, the modulus of any system is the reciprocal of the naperian logarithm of the base of the system. If in (50) a=10, we shall have the value of the modulus of the common system, or 1 M= L. 10° The naperian logarithm of 10, as will be shown hereafter, (Art. 113,) is 2.30258,50930. Therefore 1 Briggs' modulus = =.43429,44819, 2.30258,50930 which agrees with the value found by another method in Art. 93. 100. If we substitute in (49) the value of M given by (50), we have L. b (51) log. b = L. a That is, the logarithm of a number to any given base is equal to the naperian logarithm of that number divided by the naperian logarithm of the given base. * We here designate naperian logarithms by the letter L., to distinguish them from other logarithms. 101. If we make b=e in (49), we have log. e = - MX L. e= M. (52) That is, the logarithm of Napier's base in any system is equal to the modulus of that system. Hence, Briggs' modulus is the common logarithm of Napier's base, or log. 2.718281828 &c. = .4342944819 &c. 102. With the aid of these principles, we can construct as many systems of logarithms as we please after having calculated a table of naperian logarithms; for, by (49), we have only to multiply all the logs. of this table by the modulus of the new system, in order to obtain the logarithms in this system; or, by (51), we have only to divide each naperian logarithm by the naperian logarithm of the new base, to obtain the logarithms to that base. TRANSFORMATIONS OF THE LOGARITHMIC FORMULA. 103. We may now suppose the modulus to be known, since it may be determined for any base by formula (42). We are not, however, yet prepared to construct a table, for (34) is not convergent unless b is less than 2. Various transformations have been proposed, by which it is rendered very convergent and expeditious in practice. The most useful of these we proceed to explain. n n n2 d n+d 104. Since (34) will converge when b< 2, let b=1+ d whence b—1=, and log. b= log. (n+d)- log. n, (Art. 61.) n The formula becomes d d3 d4, di log. (n+d)— log. n=M at – &c.) (53) 2n? 3n3 5n5 + n 4n4 d Again, let b=1 whence 6-1 = log. (n—d)- log. n. The formula becomes n n n 5n5 Add (54) to (53); we have do d4 do log.(n+d)-2 log.n+log.(n=d)=M(- &c.) (55) &c.) 3n6 or transposing and reducing, d” do log. (n+d)=2 log.n-log:(n-d)-MC +++ &c.) + &c.) (56) Subtract (54) from (53); we have 2d 2d3 2d5 log. (n+d) – log. (n—d)=M(* + + + &c.) (57) 3n3 or transposing and reducing, ds 05 log. (n + d) = log. (n–d) +2M6 + + + &c.) (58) 3n3 5n5 d These formulæ will always converge when is a proper fraction ; that is, when d <*. In using (53) log. n must be known before log. (n+d) can be determined. From (54) we find log. n when log. (n—d) is given ; from (56) we find log. (n+d) when log. n and log. (n—d) are given ; and from (58) we find log. (n+d) when log. (n—d) is given. n 105. In (58) let n +d=p, n-d=1, whence 2n=p+1, d 2d=p-1, and ; then we have (since log. (n—d)= log. l=0) p-1 p+1 n ] (59) -1 1 log. p=2M + + p+1 + &c. 5 + This formula requires no logarithms to be given, but converges slowly unless p is a small number. 106. In formulæ (53, 54, 56, 58) let d=1; then we have In (58) substitute n+d=p+1, n-d=p, whence we have d 1 2n=2p+1, 2d=1, and ; the formula becomes 2p+1 n log:(p+1)=log-p+2M(2p+1+3(2p+1)e+5(2p+1)* +&c.) (64) 1 1 These formula will be found convenient in calculating the logarithm of a number, when that of a number greater or less by unity is given. 107. The above would suffice for the computation of a table, but other formulæ have been found, in which the series are more rapidly convergent. That of Borda may be thus derived. In (58) let n+d=(p-1)o(p+2)=p3-3p+2 nmd=(p+1)* (P—2)=p8—3p-2 whence we find n=p:— 3p, d=2, and log. (n+d)=2 log. (p-1)+log. (p+2) log. (n—d)=2 log. (p+1)+log. (P-2) These values substituted in (58) give log. (p+2) - 2 log. (p+1)+2 log. (p—1)— log. (2-2)= 1 2 + &c. (65) 5 5 By this formula we may obtain log. (p+2) when the logarithms of p+1, p-1 and p-2 are given. The series is so convergent that when p=50 the first significant figure of the second term is in the 15th place of decimals. 108. A still more convergent formula, but requiring more loga. rithms to be known, is that of HAROS. In (58) let n+d=(p—3)(2+3)(p—4)(2+4)=p4—25p+144 =p4—25p whence we have n=p- 25p+72, d=72, and the formula becomes log. (p+5) log. (p+4)— log. (P+3)+2 log. p— log. (P–3) log. (p—4) +log. (p–5)= _2M 72 1 ) p*—25p?+72 3 -25p This formula expresses the relation between seven logarithms, and the series is very convergent. 109. In (64) let p+1=q, p==q—1=(4-1)(+1); the formula becomes log. q = ] [ log. (9 — 1) + log. (q +1)] + 1 (67) 2941 3 which is a very convenient formula, requiring but two logarithms to be given. (22_1+ 110. An unlimited number of formulæ may be derived from (53) and (54) by a peculiar combination, the law of which will be perceived in the following examples. In (53) let d=3, 2, 1, and in (54) let d=1, 2, 3, successively; we shall thus obtain the six following formulæ : |