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**Unformatted text preview: **Appendix A. Exponential and Logarithmic Functions For fixed b > 1, the function b x was defined in Exercise 6 on p.22 in the textbook “Principles of Mathematical Analysis” by W. Rudin. It satisfies b x > 0, and (E1). b x + y = b x b y for real x, y . In particular, b = 1, which implies 1 = b = b y +(- y ) = b y b- y = ⇒ b- y = ( b y )- 1 = ⇒ (E2). b x- y = b x b- y = b x /b y for real x, y . (E3). ( b x ) y = b xy for real x, y . We divide the proof of this property into a few steps. Step 1 . y = n is natural. Then by iterating of (E1), ( b x ) n = b x b x ··· b x | {z } n times = b xn . Step 2 . y = − n , where n is natural. Since ( b x ) n ( b x )- n = 1, we get ( b x )- n = ( ( b x ) n )- 1 = ( b nx )- 1 = b- nx . Together with the obvious case y = 0, the cases 1 and 2 cover all integers y . Step 3 . y = 1 /n , where n is natural. We have ( e x/n ) n = b nx/n = b x = ⇒ ( b x ) 1 /n = b x/n . Step 4 . y = m/n – a rational number. Here m is integer and n is natural. Then ( b x ) y = ( b x ) m/n = ( ( b x ) 1 /n ) m = ( b x/n ) m = e mx/n = e xy ....

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