Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.3* The Natural Exponential Function - 6.3* Exercises - Page 452: 6


(a) $x=±\sqrt (e^{3}+1)$ (b) The calculated values for $x$ are ln$2$ and $0$.

Work Step by Step

(a) Apply exponential functions to both sides of the equation: $e^{ln(x^{2}-1)}=e^{3}$ $(x^{2}-1)=e^{3}$ This implies $x=±\sqrt (e^{3}+1)$ (b) Given: $e^{2x}-3e^{x}+2=0$ Suppose $e^{x} = t$ Then $t^{2}-3t+2=0$ gives $t=2,1$ Thus, $e^{x} = 2,1$ Case 1: For $e^{x} = ln2$ ln$e^{x} =2$ means $x=$ln $2$ Case 2: For $e^{x} = 1$ ln$e^{x} =1$ means $x=$ln $1$ But ln$1=0$ Thus, $x=0$ Hence, the calculated values for $x$ are ln$2$ and $0$.
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