As the difference of the errors, to the dif- So is the least error, to the correction required Ex. 1. Find the value of x in the equation x=256. Taking the logarithms of both sides (log. x) Xx=log. 256 Let x be supposed equal to 3.5, or 3.6. Then 0.09844 0.1::0.40556 0.4119, the correction. This added to 3.6, the second assumed number, makes the value of x=4.0119. To correct this farther, suppose x=4.011, or 4.012. By the first supposition. x=4.011, and log.x=0.60325 x=4.012, and log.x=0.60336 By the second supposition. Multiplying by 4.012 Then 0.00105 0.001::0.01139 0.011 very nearly. Subtracting this correction from the first assumed number 4.011, we have the value of x=4, which satisfies the conditions of the proposed equation; for 41=256. 2. Reduce the equation 4x=100x3. 3. Reduce the equation x=9x. Ans. x=5. 64. The exponent of a power may be itself a power, as in the equation 22 am = b; where x is the exponent of the power m", which is the expo Ex. 4. Find the value of x, in the equation 93 =1000. 3 X (log. 9)=log. 1000. Therefore 3*= Then as 3*3.14. x(log. 3)=log. 3.14. log. 1000 log. 9 3.14. In cases like this, where the factors, divisors, &c. are logarithms, the calculation may be facilitated, by taking the logarithms of the logarithms. Thus the value of the fraction 41 is most easily found, by subtracting the logarithm of the logarithm which constitutes the denominator, from the logarithm of that which forms the numerator. 4969296 42 SECTION IV. DIFFERENT SYSTEMS OF LOGARITHMS, AND COMPUTATION OF THE TABLES. 65. For the common purposes of numerical computation, Briggs' system of logarithms has a decided advantage over every other. But the theory of logarithms is an important instrument of investigation, in the higher departments of mathematical science. In its numerous applications, there is frequent occasion to compare the common system with others; especially with that which was adopted, by the celebrated inventor of logarithms, Lord Napier. In conducting these investigations, it is often expedient to express the logarithm of a number, in the form of a series. If a=N, then x is the logarithm of N. (Art. 2.) To find the value of x, in a series, let the quantities a and N be put into the form of a binomial, by making a=1+b, and N=1+n. Then (1+b)=1+n, and extracting the root y of both sides, we have (1+b);=(1+n)} By the binomial theorem b2 (1+6);=1+2 (6) + " ( ~; − 1 ) ( )+C-1) C−2) 1b2 2.3)+&c. y 2 y n3 (13) + &c. 2.3 As these expressions will be the same, whatever be the value of y, let y be taken indefinitely great; then and being indefinitely small, in comparison with the numbers - 1, -2, &c. with which they are connected, may be cancelled from the factors (-1)-2), &c. (-1). (-2), &c. y 2 (Alg. 456.) leaving 1+() + ( ) = ( ) &c. Rejecting 1 from each side of the equation, multiplying by y, (Alg. 159.) and dividing by the compound factor into which is multiplied, we have x=Log. N="n2+}n3¬ }n1+ &c. 3 4 b − 1 b2 + ¦ b 3 — 1b1+ &c. Or, as n N-1, and b-a-1, = 2 − A Log. N=(N-1) — 3(N − 1 ) ' + ¦ (N − 1 ) 3 — ¦ (N − 1 ) '+&c. (a-1)-(a-1)2 + (a−1)-(a− 1)+&c. 2 3 Which is a general expression, for the logarithm of any number N, in any system in which the base is a. The numerator is expressed in terms of N only; and the denominator in terms of a only: So that, whatever be the number, the denominator will remain the same, unless the base is changed. The reciprocal of this constant denominator, viz. 1 (a−1)-(a−1 ) 2 + ¦ ( a − 1 ) 3 — ¦ ( a − 1 ) 1 + &c. is called the Modulus of the system of which a is the base. If this be denoted by M, then Log. N=M×((N−1)—¿(N − 1 )2 + } (N − 1 ) 3 — ↓ (N − 1)a +&c. ) 66. The foundation of Napier's system of Logarithms is laid, by making the modulus equal to unity. From this condition the base is determined. Taking the equation above marked A. and making the denominator equal to 1, we have x = n = {n2 + {} n 3 — } n 1 + } n 5 - &c. By reverting this equation* 2 2.3 2.3.4 2.3.4.5 Or, as by the notation, n+1=N=a*, a2=1+x+* +: + + If then be taken: equal to 1, we have 1 1 1 a=1+1++ +. + 1 + &c. 2 2.3 2.3.4 2.3.4.5 Adding the first filcen terms, we have 2.7182818284 Which is the base of Napier's system, correct to ten places of decimals. * See note D. Napier's logarithms are also called hyperbolic logarithms, from certain relations which they have to the spaces between the asymptotes and the curve of an hyperbola; although these relations are not, in fact, peculiar to Napier's system. 67. The logarithms of different systems are compared with each other, by means of the modulus. As in the series. (N − 1) — ¦(N − 1 )2 + (N − 1 ) 3 — ¦ (N − 1 )1 +&c. 2 3 (a− 1) — (a− 1 )2 + ¦ (a− 1 ) 3 — (a−1)1+&c. which expresses the logarithm of N, the denominator only is affected by a change of the base a; and as the value of fractions, whose numerators are given, are reciprocally as their denominators: (Alg. 360. cor. 2.) The logarithm of a given number, in one system, Is to the logarithm of the same number in another system; To the modulus of the other. So that, if the modulus of each of the systems be given, and the logarithm of any number be calculated in one of the systems; the logarithm of the same number in the other system may be calculated by a simple proportion. Thus if M be the modulus in Briggs' system, and M' the modulus in Napier's; the logarithm of a number in the former, and l' the logarithm of the same number in the latter; then, M : M'::7: 7', Therefore, 17'x M; that is, the common logarithm of a number, is equal to Napier's logarithm of the same, multiplied into the modulus of the common system. To find this modulus, let a be the base of Briggs' system, and e the base of Napier's; and let la denote the common logarithm of a, and l'.a denote Napier's logarithm of a. Then M 1::l.a: l'.a. Therefore M-1.a l'.a But in the common system, a=10, and l.a=1. 1 So that, M= that is, the modulus of Briggs' system, 1.10' is equal to 1 divided by Napier's logarithm of 10. |