Therefore, replacing x by x log, a, and remembering that This Exponential Series is convergent for all finite values of x. The Logarithmic Series. 6. From the expansion of a in Art. 5, we have The series in the second member is convergent for all finite values of x. Therefore, by Ch. XXXV., § 1, Art. 3, its sum approaches loga, as a=0. 7. Substituting 1+x for a in the relation of the preceding article, we have a (α- 1) If <1 numerically, the expansion of (1 + x)a by the Binomial Theorem is a convergent series. Therefore, (1 + x)=1+ ax + a (x-1) (-2) 203 +·· 3 whence, loge (1 + x) = x (2) It is important to keep in mind that this expansion holds only when a lies between - 1 and +1. It is therefore of little practical value in computing logarithms. A series which can be used for this purpose will be derived in the next article. loge(1+n)=logen+2| loge 2n+1 3(2n+1)3 ' 5(2n+1)3 1 1 1 + + + ...] 1 + (3) + 1 The series (2) is convergent when <1, and positive. Therefore the series (3) is convergent when 1 2n+1 <1; or n>o. Hence, this series is equal to log.(1+n) for all positive values of n, however great. Computation of Logarithms. 9. Naperian Logarithms. By means of the formula derived in the preceding article, the naperian logarithms of all positive numbers can be computed. = 0 + 2 (.3333333 + .0123457 + .0008231 + .0000653 = +.0000057 + .0000005) = .69315, to five places of decimals. It is necessary to compute only logarithms of prime numbers, by the formula. Logarithms of composite numbers are obtained by adding together the logarithms of their factors. As the numbers increase, fewer terms in the formula are required to compute their logarithms to a given decimal place. 10. Common Logarithms. · Let n = logo N, or 10" = N. Taking logarithms, to base e, n log, 10 log, N. is called the Modulus of the common system with respect to the Naperian system. Hence, to compute the common logarithm of any positive. number, multiply the naperian logarithm by the modulus of the common system, .43429. E.g., log10 2 = .43429 × log, 2, = .43429 × .69315, = = = .30103. Only common logarithms of prime numbers should be thus obtained. Common logarithms of composite numbers are obtained by adding together the logarithms of their factors. 11. Evidently, the naperian logarithm of a number can be obtained from the common logarithm by dividing by .43429, or by multiplying by 2.30259. The number 2.30259 is called the Modulus of the naperian system with respect to the common system. EXERCISES. 1-10. Compute the common logarithms of all integers from 11 to 20 inclusive. have evidently the solution 0, 0. Let us inquire if they can be simultaneously satisfied by values of x and y other than 0, 0. Multiplying (1) by b2, and (2) by ɑ2, Subtracting, if This equation will be satisfied by a value of x other than 0, The same result would have been obtained, had we first eliminated y. is an example of a form which occurs frequently in mathematics, and for which a special notation has been devised. Such an expression is called a Determinant. The determinant (I.) is usually written in a square form: Ե, Ն. The positive term of the determinant, obtained from the square form, is the cross-product from upper left to lower right; the negative term is the cross-product from upper right to lower left. |