Now log xy log x log y and so log x 2 2 log x log x 3

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Now log xy = log x + log y and so log x 2 = 2 log x, log x 3 = 3 log x, . . . , log x n = n log x, where n is any positive integer. Hence log e n = n log e = n. Again, if p and q are any positive integers, and e p/q denotes the positive q th root of e p , we have p = log e p = log( e p/q ) q = q log e p/q , so that log e p/q = p/q . Thus, if y has any positive rational value, and e y denotes the positive y th power of e , we have log e y = y, (1) and log e - y = - log e y = - y . Hence the equation (1) is true for all rational values of y , positive or negative. In other words the equations y = log x, x = e y (2)
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[IX : 204] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 453 are consequences of one another so long as y is rational and e y has its positive value. At present we have not given any definition of a power such as e y in which the index is irrational, and the function e y is defined for rational values of y only. Example. Prove that 2 < e < 3. [In the first place it is evident that Z 2 1 dt t < 1 , and so 2 < e . Also Z 3 1 dt t = Z 2 1 dt t + Z 3 2 dt t = Z 1 0 du 2 - u + Z 1 0 du 2 + u = 4 Z 1 0 du 4 - u 2 > 1 , so that e < 3.] 204. The exponential function. We now define the exponential function e y for all real values of y as the inverse of the logarithmic function. In other words we write x = e y if y = log x . We saw that, as x varies from 0 towards , y increases steadily, in the stricter sense, from -∞ towards . Thus to one value of x corresponds one value of y , and conversely. Also y is a continuous function of x , and it follows from § 109 that x is likewise a continuous function of y . It is easy to give a direct proof of the continuity of the exponential function. For if x = e y and x + ξ = e y + η then η = Z x + ξ x dt t . Thus | η | is greater than ξ/ ( x + ξ ) if ξ > 0, and than | ξ | /x if ξ < 0; and if η is very small ξ must also be very small. Thus e y is a positive and continuous function of y which increases steadily from 0 towards as y increases from -∞ towards . More- over e y is the positive y th power of the number e , in accordance with
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[IX : 205] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 454 the elementary definitions, whenever y is a rational number. In particular e y = 1 when y = 0. The general form of the graph of e y is as shown in Fig. 53 . 0 X Y 1 Fig. 53. 205. The principal properties of the exponential function. (1) If x = e y , so that y = log x , then dy/dx = 1 /x and dx dy = x = e y . Thus the derivative of the exponential function is equal to the function itself . More generally, if x = e ay then dx/dy = ae ay . (2) The exponential function satisfies the functional equation f ( y + z ) = f ( y ) f ( z ) . This follows, when y and z are rational, from the ordinary rules of indices. If y or z , or both, are irrational then we can choose two sequences y 1 , y 2 , . . . , y n , . . . and z 1 , z 2 , . . . , z n , . . . of rational numbers such that lim y n = y , lim z n = z . Then, since the exponential function is continuous, we have e y × e z = lim e y n × lim e z n = lim e y n + z n = e y + z .
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