2 log.( 1 1 1 1 1 + + + + &c.) (8) 3p3 5p 7p 9p99 As p may be any positive number, greater than 1, make Then p=2x+1, and equation (8) becomes By this last equation we perceive that the logarithm of (z+1) will become known when the log. of z is known, and some value assigned for the constant M. coverer of logarithms, gave M the the sake of convenience. Baron Napier, the first disarbitrary value of unity, for Then, as in every system of logarithms, the logarithm of 1 is 0, make z=1, in equation (9), and we shall have 1 1 1 log. 2=2 ( 3+3 (3+5/3+, &c.) =.0.69314718. st, This is called the Napierian logarithm of 2, because its magnitude depends on Napier's base, or on the particular value of M being unity. Having now the Napierian logarithm of 2, equation (9) will give us that of 3. Double the log. of 2 will give the logarithm of 4. Then, with the log. of 4, equation (9) will give the logarithm of 5, and the log. of 5 added to the log. of 2, will give the logarithm of 10. Thus the Napierian logarithm of 10 has been found to great exactness, and is 2.302585093. The Napierian logarithms are not convenient for arithmetical computation, and Mr. Briggs converted them into the common logarithms, of which the base is made equal to 10. To convert logarithms from one system into another, we may proceed as follows: Let e represent the Napierian base, and a the base of the common system, and Nany number. Also, let x represent the logarithm of N, corresponding to the base a, and y the logarithm of N, corresponding to the base e. Now, by inspecting these equations, it is apparent that if the base a is greater than the base e, the log. x will be less than the log. y. Taking the logarithms of both members, observing that x and y are logarithms already, we have x log. a=ylog.e. This equation is true, whether we consider the logarithms taken on the one base or on the other. Conceive them taken on the common base, then In this equation x and y must be logarithms of the same number, and therefore if we take x=1, which is the logarithm of 10, in the common system, y must be 2.302585093, as previously determined. This last decimal is called the modulus of the common system; for by equation (10) we perceive that it is the constant multiplier to convert Napierian or hyperbolic logarithms into common logarithms. But equation (9) gives Napierian logarithms when M=1; therefore the same equation will give the common logarithms by causing M to disappear, and putting in this decimal as a factor. Equation (9) becomes the following formula for computing ommon logarithms: 0.86858896 1 log.(z+1)-log.z= 1 (2= +1+3(2x+1)3 +5(2x+1)3+, &c.)(F) To apply this formula, assume z=10. Then If we make z=99, then (z+1)=100, and log.(z+1)=2, 2.00000-log.99=0.00436481 Hence By transposition Subtract log.99=1.99563519 log.11=1.04139269=log.11 log. 9-0.95424234 log.9=log. 3=0.47712117=log.3. Thus we may compute logarithms with great accuracy and rapidity, using the formula for the prime numbers only. By equation (11) we perceive that the logarithm of the Napierian base is 0.434294482; and this logarithm corresponds to the number 2.7182818, which must be the base itself. We may also determine this base directly: In the fundamental equation (1), the base is represented by (1+c). In equation (A), c must be taken of such a value as c2 C3 C4 shall make the series c- + +, &c., equal to 1. 2 3 4 determine what that value shall be, in the first place, put Now by reverting the series (Art. B.), we find that But to But, by hypothesis, the series involving c equals unity, that is, y=1. Therefore By taking 12 terms of this series, we find (1+c)=2.7182818, the same as before. If we take equation (3), making M=1, we find the Napierian This series will not converge rapidly unless p is a large numBut by equation (2) x=log. (p+1)-log. p. If in this equation we make p=1, we shall find the Napierian log. of 2 equal to the series But we have already found the same logarithms equal to the series 1 2 (+3/+3/+, &c.); therefore these two series 3(5)&t, are equal to each other, and because the former did not rapidly converge, it became necessary to obtain the latter USE AND APPLICATION OF LOGARITHMS. (Art. 152.) The sciences of trigonometry, mensuration, and astronomy alone, can develop the entire practical utility of logarithms. The science of algebra can only point out their nature, and the first principles on which they are founded. To explain their utility, we must suppose a table of logarithms formed, corresponding to all possible numbers, and by them we may resolve such equations as the following: 1. Given 2=10 to find the value of x. If the two members of the equation are equal, the logarithms of the two members will be equal, therefore take the logarithm of each member; but as x is a logarithm already, we shall have x log. 2=log. 10. 2. Given (729) * =3, to find the value of x. Raise both members to the x power, and 3-729=93, 3. Given a+b=c, and a¤—by=d, to find the values of x By subtracting the second equation from the first and making c—d=2n, we shall find y=log. a 3 4. Given (216)*=12, to find the value of x. log. n |