Let us, for instance, seek the 11th root of 2044. Here 211 2048, and hence we have By calculating only the terms here written down, we obtain 11/2044 = 1.999644570706, true, probably, within one unit in the last figure. When, however, the quantity v is nearly = 1, or indeed above 5, the convergency becomes very slow; as it is obvious that the successive factors of the several coefficients continually increase and approach towards unity as their common limit, in all the roots. The convergency depends then on the smallness of v, which causes its powers to diminish rapidly in value. This difficulty, however, may be completely evaded by taking two figures instead of one for the first approximation. It will be well to take 4, 5, 6, or 7, for the second figure, according as the first taken with it shall form a number decomposable into factors never greater than 12, and such as shall be supposed most likely to approximate closely to the true root. Thus, for instance, the square root of 6 gives, whilst we take only one figure, 2 (1 + 1), or 3 (1—})2 which would converge slowly: but if we take two figures, as 2·5, we have 2.5 (1 ), which converges very rapidly. And, in all cases, the binomial theorem enables us to secure this rapid convergency *. Ex. Let the student extract the roots which follow by this method. 3/7; 3/9; √17; 5/246; and calculate to six decimals the values of the following binomial surds: 3+109; —7/006564; and √ — 1 × 3⁄4√√ 16. THE EXPONENTIAL THEOREM. The expression a takes the form of a simple term: but it is of great importance to develop it in a series proceeding according to powers of x, as in the last That is, to find the coefficients of the series in case. (3) a*+ = A。 + A‚(x + y) + A2(x + y)2 + A ̧(x + y)3 + ... Multiply (1) (2) together, and equate it to (3): then equating the coefficients Again equating the homologous coefficients of x in this, we have the results *This is the method most commonly employed by foreign mathematicians for approximating to the roots of numbers, when more figures are required than can be obtained by the logarithmic tables. Bourdon, Algèbre, p. 290. Whence, omitting the subscribed accent from A1, the development is It still remains to determine the value of A in terms of a, which may be thus effected. Put a = {1 + (a− 1)}*; then expanding the binomial, we have Now the coefficients of a1 in the several terms are as follows: (a 1)3 + .... Whence A = (a — 1) — — (a — 1)2 + }.(a — 1)3 — † (a — 1)1 + For the purposes of calculation, this expression is generally useless, on account of its want of convergency. As an analytical expression, however, it is an éssential element in the deduction of the formula for logarithms; and the necessity of its calculation here is avoided by taking a different subsequent course. LOGARITHMS. I. DEFINITIONS AND ELEMENTARY PROPERTIES. = IN the equation a* N, a is called the base of the system, N the number, and the logarithm of N to the base a. This is generally denoted by the equation, x = log,N, or x = 1,N, where the base of the system is written as a subscribed letter to the contractions "log" or "l" of the word logarithm. Logarithms are said to be of different systems, according to the value of the base a. As logarithms are, by the definition, only indices of the powers of the base a, it will be obvious that the fundamental operations will be the same as those of indices already explained. It will, nevertheless, be advantageous to collect into one place, and with appropriate phraseology, the simple propositions relative to these indices which we shall have occasion to employ. They are, in fact, the rules for the use of logarithms; and the only difficulty in the inquiry is the actual calculation of the logarithms themselves. 1. The sum of the logarithms of two numbers is equal to the logarithm of their product. For let a N, and a = N1: then a*a" = a*+ = N N1, or x + y = log, N N1. 2. The difference of the logarithms of two numbers is equal to the logarithm of their quotient. 3. The logarithm of the nth power of any number is equal to n times the logarithm of that number. 4. The logarithm of the nth root of any number is the nth part of the logarithm of that number. 5. If a series of numbers be taken in geometrical progression, their logarithms are in arithmetical progression. For any number may be represented by a". Let a" be the first term of the geometrical series, and a" the ratio: then the series are For the numbers a", a+n aTM+2", m+ 3n and for the logs. m, m + n, m + 2n, m + 3n and it is obvious that these logarithms are in arithmetical progression, whatever the base of the system may be. 6. The logarithm of the base in every system is 1. For a1a, or log, a = 1. 7. The logarithm of 1 in every system is 0. For ao 1, or log, 1 = 0. 8. If a table of logarithms be calculated to any one system, those for another given system can be obtained from these by the use of a constant multiplier for all the logarithms of the first table. Whence, taking log of N in both systems, we have log, N = x, 1 and log.N = Mx, where M depends upon the bases a, and a, and is constant for all values of x, so long as the systems remain the same. It will therefore follow, that if we can more easily compute logarithms to one base a,, than to any other a, we may avail ourselves of it, and convert them to another system by means of the proper multiplier M. 9. As a general mode of finding Ma, we have, from the last equation, Whence, if we can compute the logs. of any one number N in the two systems, we can obtain the requisite multiplier for all the other transformations. The number M is called the modulus of the system of logarithms; and refering to the base a, it is written M., signifying the modulus to the base a. 10. If a, b be the bases of two systems, and N, N, any numbers whatever : loga N log, N then = log. N1 log, Ni For let e be the base whose modulus is unity: then we have log NM log, N, log, N, M, log, N,, log,N= M, log,N, a II. LOGARITHMIC SERIES. In the equation a*N, to find an expression for the value of x in terms of a and N. Raise both sides to the zth power; then we have a** N*. Develop both sides by the exponential theorem: then we obtain Axz A2x2z2 1 + + in which A=(a-1)-(a—1)2+(a−1)3—...; and A, (N-1)-(N-1)2+(N-1)3—... A ̧=(N—1)—}(N−1)2+}(N—1)3—... Equating the homologous coefficients of the indeterminate quantity z, we have from any one of the resulting equations, as that of z1, for instance, A1 (N − 1) — 1 (N − 1)2 + } (N · Ax= · 1)3 A1, or x = A (a — 1) — — (a 1)2 + } (a — 1)3 . which is an expression for x, the logarithm of N to the base a. It is more usual to write n instead of N a 1, and M. instead of and this reduces the expression to x= log. (1 + n) = M. {n — n2 + } n3 — ‡ n2 + .... } This series is not in a form well adapted for calculation, except when n is a small fraction. The following process will transform it into another adapted to any number whatever. Substitute n for n, and write the two equations, log. (1+n) M1 { + n − } n2 + } n3 — ↓ n1 + .... } log. (1-n) = M1{+ = Hence by subtraction, = 2M. {n + n3 +225 + .... } p+1 Ρ a + 2p+1 3(2p+1)3 + 5(2p+1) + ...} =log. (p+1)-log.p=2M or finally, log. (p+1)= log. p+2M. 1 (2p+1 3(2p+1)3 + + Hence, whenever we can calculate log.p, we can, by means of this series, calculate log.(p+1); and the series converges the more rapidly as p becomes greater.* * Several improvements of this formula, at least in respect of practical application, have been proposed by different writers; but as the tables have already been computed and verified, they are, in this point of view, of little importance. Nevertheless, it may not be out of place to merely indicate one or two of them, referring for more ample details to the elegant little treatise of Professor Young, on the "Computation of Logarithms," second edition, 1835. +2 log (p+1) −2 log (p−1)+2M {μ323p+3 + 2 5 + p3-3p ....} 2. Put III. ON THE COMPUTATION OF LOGARITHMS. 1. To find the value e of a which will render M, =1= or A = 1. A, or A =1= M., and A = 1. + .... Computing this series to thirteen terms, we have e = 2.718281828 ...... Logarithms calculated for the base e, or modulus 1, are called napierean, from their inventor Lord Napier. They are also often called hyperbolic logarithms, from an analogy which exists between them and the spaces contained by the rectangular hyperbola and its asymptotes. Under the latter name they are given in Hutton's Tables. 2. To calculate the hyperbolic logarithms. The general series for the logarithm of p + 1 is, in this case, Log. (p + 1) = log. p + 2 1 1 + + 2p+1 3(2p+ 1)3 Now, (theor. 7, p. 249) log. 1 = 0, and hence And in the same way, the series may be continued to any extent required. the logarithms of numbers which are either products or quotients, powers or m 2. Put for p in the equation of the text; and then if we make m =x^ — 25x2 and m +n = xa— 25x2 + 144, we get log (x+5) = log (x+3)+ log (x−3) + log (x+4) + log (x—4) 3. Put P = 1+n In -2 M { which is the then we get n = P. Put also p⇒xo — 98x1 + 2401æ2, and q=x¤ — 98x42401x2-14400; from which is obtained the formula of Lavernède, 2 log x + 2 log (x+7)+2log (x—7)—log (x+8)—log (x-8) — log (x+5) — log (x-5) — log (x+3)— log (x—3) =2M { ¿6 — 98x+ + 2401.a2 — 7200 + · 98a+ + 2401.æ2 — 7200) + } |