be the required quotient. Therefore, by a simple subtraction we obtain the quotient of a division. 216. Formation of Powers, and Extraction of Roots. Let it be required to raise a number y to a power denoted by m n ; a denoting the base, and x the logarithm of y, we have the equation y=a* ; that is, the logarithm of any power of a number is equal to the product of the logarithm of the number, by the exponent of the power. Take n=1, as a particular case; there will result. . log y"=m. log y, an equation which is susceptible of the above enunciation Let m =1; there will result 1 y= log y or log Vy log y ; that is, the logarithm of any root of a number is equal to the logarithm of this number, divided by the index of the root. Consequence. To form any power of a number, take the logarithm of this number from the tables, multiply it by the exponent of the number, then find the number corresponding to this product, it will be the required power. In like manner, to extract the root of a number, divide the logarithm of the proposed number by the index of the root, then find the number corresponding to the quotient, it will be the required root. Therefore, by a simple multiplication, we can - raise a quantity to a power, and extract its root by a simple division. 217. The properties just demonstrated are independent of any system of logarithms; but the consequences which have been deduced from them, that is, the use that may be made of them in numerical calculations, supposes the construction of a table, containing all the numbers in one column, and the loga-rithms of these numbers in another, calculated from a given base. Now, in calculating this table, it is necessary, in considering the equation ay, to make y pass through all possible states of magnitude, and determine the value of x corresponding to each of the values of y, by the method of No. 209. The tables in common use, are those of which the base is 10, and their construction is reduced to the resolution of the equation 10=y. Making in this equation, y successively equal to the series of natural numbers, 1, 2, 3, 4, 5, 6, 7,-, we have to resolve the equations 10=1, 102, 10'=3, 10:=4 We will moreover observe, that it is only necessary to calculate directly (by the method of No. 209) the logarithms of the prime numbers 1, 2, 3, 5, 7, 11, 13, 17, - - - -; for as all the other entire numbers result from the multiplication of these factors, their logarithms may be obtained (No, 215) by the addition of the logarithms of the prime numbers. Thus, since 6 can be decomposed into 2×3, we have log 6 log 2+log 3, in like manner. 24-23×3; A gain hence log 24=3 log 2 + log 3. 360 23 X3 X5; hence log 360-3 log 2+2 log 3+ log 5. It is only necessary to place the logarithms of the entire numbers in the tables; for, by the property of division (No. 215) we obtain the logarithm of a fraction by subtracting the logarithm of the divisor from that of the dividend. 218. If we had a table of logarithms constructed, it would be easy to construct from this as many as we wished. For let a be the base of a system already formed, and b be the base of a system which it is required to construct; let N represent any number whatever, and log N and X, its two logarithms calculated from the bases a and b; we have the equa X tion b = N. Whence taking the logarithms of both members, in the system of which the base is a, X. log blog N. This proves that, knowing the logarithm of a number in one system, in order to have the logarithm of the same number in another system, we must divide the logarithm of the number calculated in the first system, by the logarithm of the base of the second system, also calculated in the first system. Thus the logarithm of 4, in the system of which the base is log 4 3, is log 3, log 4 and log 3 being calculated in the known sys 3' tem of which the base is 10. Let N, N', N" ---- be a series of numbers, a the base of a system already formed, b that of a system to be constructed, we have the equations 1 1 log N; X'= .log N'; X"= log N" log b log b ; from which we see that, a table being already formed, in order to construct a new one from it, we must multiply the logarithms of the first system by the constant quantity 1 log b This constant quantity which serves to pass from one table to another, is called the modulus of the new table, with reference to the old. § III. Logarithmic and Exponential Series. The method of resolving the equation ab, exposed in No. 209, is sufficient to give an idea of the construction of logarithmic tables; but this method is very laborious when we wish to approximate very near the value of x. Analysts have discovered much more expeditious methods for constructing new tables, or for verifying those already calculated. These methods consist in the development of logarithms into series. 231. Let it be required to develope a number, represented by y, into a series, and apply the method of indeterminate coefficients (No. 189). It is immediately visible that we cannot suppose ly = A + By + Cy2 + Dy3 + &c. ; for making y0, the first member reduces (219) to an infinite regative, or an infinito positive quantity, according as the base is greater or less than 1; whilst the second member would be reduced to A. Neither can we suppose l. y=Ay+By2+&c., but if we put y under the form 1+x, and suppose 1. (1+x)=Ax+Bx2 +Сx3 +Dx2 + - - - - (1), making x=0, the equation is reduced to 1. 1=0, which does not present any absurdity. In order to determine the coefficients A, B, C...., we will follow the process of No. 190. Substituting z for x, the equation becomes 1. (1+z)=Az+Bz2 + Cz3 + Dz2 + - - - Subtracting the equation (2) from (1), we obtain (2). l. (1+x)−l.(1+z)=A(x−z)+B(x2 −z2)+C(x3 −z3)+...(3). The second member of this equation is divisible by x-z; we will see if we can by any artifice put the first under such a form that it shall also be divisible by x-z. We have 1. (1+x)—l. (1+z)=1. 1+x =1.(1+ X- ; X-Z but since 1+z can be regarded as a single number u, we can develope 1.(1+u), or 1. (1 + 2), in the same manner as... Substituting this development for 1.(1+x)—l.(1+z) in the equation (3), and dividing both members by x-z, it becomes 1 A. +B (x-2)2 +C +.... 3 1+z (1+z)2 (1 + z)3 =A+B(x+2)+C(x2+xz+z2)+.. Since this equation, like the preceding, must be verified by any values of x and z, make x=z, and there will res Whence, clearing the fraction, and transposing 0=A+2B x+3C x2 + 4D | x3+5E -A+ A +2B +3C + 4D Putting the coefficients of the different powers of x equal to zero, we obtain the series of equations A-A=0, 2B+A=0, 3C+2B=0, 4D+3C=0 --- ; The law of the series is evident; the coefficient of the nth term A is equal to , according as n is even or odd; hence we will obtain for the development of 1.(1+x), N. B. By the above method, the coefficients B, C, D, E.... have all been determined in functions of A; but A remains entirely undetermined. Now this should be the case from the nature of the expression which it has been proposed to develope; for since an infinite number of systems of logarithms can be formed, the general development of 1. (1+x) must necessarily involve an indeterminate quantity, which serves to distinguish the systems from each other. Moreover, we have seen (218) that the logarithms of the same number, taken in two systems, only differ by a factor, which is the same for all numbers; therefore the indeterminate quantity ought to be a factor of the series. The number A is (218), the modulus, the particular value of which characterizes the system of logarithms. 232. The most simple hypothesis that can be made is A= 1, which gives denoting this particular system of logarithms by l'.(1+x). |