THEOREMS OF EXPANSION.] . [ * ror log (1 +x) = A I x
(x 2 - x - - &c. ALGEBRA 557 Now log (!+#) log (1 a;) = log > and x has to be found, so that shall be the quotient of l + x consecutive numbers = This gives , , l + u and log l+u u x = Zu+l 1 127. To apply this formula to the calculation of com mon logarithms, we will commence by finding from it a few logarithms of the system for which A= 1. In this system 1. H.-l f kgS-{| + i.i + &a} = . 693, 147, 2. 2- If w = 4, log 5 = log 4 + 2 + - ^+&c. 1 7 O O I Q *5 O-3 = 1 . 609, 437, 9 . Hence log 10 = log 2 + log 5 = 2.302, 585, 1. This system, for which A = 1 is the so-called Napierian system, which assumes no base, but defines a logarithm to be such that the increment of the number shall be the product of the number by the increment of the logarithm. In this system the number of which the logarithm is 1 is 2.718.... and is generally designated by the letter e. To pass from Napierian logarithms to common, we observe that if e* 1 Q y = n ; x is the logarithm of n to the base e, and y to the base 10. Now, taking the Napierian logarithm of each side of this equation, we obtain x = y Nap. log 10. Or ^ ~ Nap. log 10 2 . 302, 585, 1 = xx .4342944819 = #x . 4343 very nearly. This multiplier^ which was previously denoted by A, is called the modulus of the common system of logarithms. A celebrated calculator of the last century, Mr A. Sharp, found it to be 43429448190325182765112891891660508229439700 5803666566114454. For further details on the construction and use of loga rithmic tables, the reader is referred to the Article on LOGARITHMS. 3. Exponential Theorem, 128. It is now required to expand a* in terms of x. I . Write 1 + a - 1 for a, and apply the binomial theorem ; the result is Here the only term which does not contain x is 1 ; and the coefficient of x being traced through the different terms, is easily seen to be Thus will seem (Art. 126) to be the reciprocal of the modulus ol the system of logarithms whose base is a : call it r. We have now to determine B, C, &c., in term/; of r, from the form of expansion a = l+rjc + Ex 2 + O 3 + &c. Write y + z is place of x ; then IL 2 +&c. -l+ry + rz + &c. a?+ = a J xa =(1 + n/+Bi/ 2 +&c.) x(l +rz + Bz 2 +&c.) = 1 +ry + Bi/ 2 +&c. + rz + r 2 yz + rBy^z + &c. + &c. Equating coefficients of z, yz, y^z, &c., in I. and 11., we get r r. B = = r 2 , 3C = rB, ., c = and , a* = 1 + rx + 1.2 1.2.3 r 2 x 2 rV , &c + &C. 1.2 1.2.3 129. Now, since e is such (Art. 127) that e -I - ^(e-l) 2 + &c. = 1, and r = a- 1 - 1 (a - l) 2 + &c., what ever a be, it follows that when e takes the place of a, r becomes 1. O! 8 and putting Again but since, we have 1.2 x=l, e=l + 1 + 1.2.3 1 + &C. 1.2 = 2.718281828459045., e r =l+r a = 1 + r 1.2 ?- 2 2 TT2 a + &C. + &C. From this equation we have r = Nap. log a, a result obtained before. 130. We may approximate directly to the value of r when a = 1 0, thus - ) =ar* = I-rx+ , a) 1.2 - &c. so that the coefficient of x in { - Now is -r. = (! + --!), whence (Art. 128) the cooffi a I a I
- ient of x in ( - ) is
1 , 1/1 , l/i -1 -- --1 +o -- 1 ~ &c - a 2 a J 3 a / 9 1/9 10 2 2. 302.. | 1-J f&C. + &C. Additional Examples. as m becomes larger and larger. By the binomial theorem t ljich (1-i ) approacb.33
m/ ri
