By C. W. Evans (auth.)
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Extra resources for Engineering Mathematics: A Programmed Approach
It is particularly important to realize that the laws of indices 1 a ras = a r+ s 2 (ary = a r, hold whenever a > O. Note that we do not define a r when a < 0 unless r is an integer. This is to avoid violating the laws of indices. o Using a calculator and employing six successive approximations to n, Example obtain six successive approximations to 2'"'. 14159 ... 821. 825 • LOGARITHMS From the laws of indices we obtain the laws of logarithms. Logarithms are important because they provide a transformation which enables the arithmetical processes of multiplication and division to be replaced by those of addition and subtraction.
In fact we shall see later that (1 + X)" = i (fl)xr r=O r is valid when n is any real number provided -1 < x < 1. However, until we consider the concept of infinite series (Chapter 9), the expression on the right will have no meaning for us. Here is an example which we shall solve together. It shows one practical application of some of the things we have been doing. 8 Practical _ _ _ _ _ _ _ __ t ELECTRICAL FORCE x A magnetic pole, distance x from the plane of a coil of radius a, and on the axis of the coil, is subject to a force F = 7 (a- kx 7 'in r)'- + (k constant) 23 24 NUMBERS AND LOGARITHMS when a current flows in the coil.
2 ~ [1 - ~ ~ x-l a2 ] 2 x2 SUMMARY neglecting tcrms in alx of degree higher than 2. 5... _ 5ka 2 2x 6 X4 SUMMARY NUMBERS We have seen that numbers can be classified into the following subsets: natural numbers N integers 71. cQclRcC INDICES We have defined what is meant by a' and obtained the laws of indices: aras = a r+ s (a')\ = a ro- o o LOGARITHMS From the laws of indices we have deduced the laws of logarithms: loge (ab) = loge a + loge b loge (alb) = loge a - loge b loge (a r ) = r loge a log" b = loge bllog e a (a 1) o o o o '* THE BINOMIAL EXPANSION If II is any natural number, (1 + x)" = L II r~() (n) xr r 25 26 NUMBERS AND LOGARITHMS EXERCISES (for answers see p.