Chapter 6 | Applications of Integration
727
Definition For any real number x , define y = e x to be the number for which
ln y = ln( e x ) = x .
(6.25)
Then we have e x =exp( x ) for all x , and thus
(6.26)
e ln x = x for x >0and ln( e x ) = x
for all x . Properties of the Exponential Function Since the exponential function was defined in terms of an inverse function, and not in terms of a power of e , we must verify that the usual laws of exponents hold for the function e x .
Theorem 6.19: Properties of the Exponential Function If p and q are any real numbers and r is a rational number, then i. e p e q = e p + q ii. e p e q = e p − q iii. ( e p ) r = e pr
Proof Note that if p and q are rational, the properties hold. However, if p or q are irrational, we must apply the inverse function definition of e x and verify the properties. Only the first property is verified here; the other two are left to you. We have ln( e p e q ) = ln( e p )+ln( e q ) = p + q = ln ⎛ ⎝ e p + q ⎞ ⎠ . Since ln x is one-to-one, then e p e q = e p + q . □ As with part iv. of the logarithm properties, we can extend property iii. to irrational values of r , and we do so by the end of the section. We also want to verify the differentiation formula for the function y = e x . To do this, we need to use implicit differentiation. Let y = e x . Then ln y = x d dx ln y = d dx x 1 y dy dx = 1 dy dx = y . Thus, we see
Made with FlippingBook - professional solution for displaying marketing and sales documents online