A - Appendix A Exponential and Logarithmic Functions For...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

This document was uploaded on 02/15/2012.

Page1 / 3

A - Appendix A Exponential and Logarithmic Functions For...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online