Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 12 - Exponential Functions and Logarithmic Functions - 12.6 Solving Exponential Equations and Logarithmic Equations - 12.6 Exercise Set: 27

Answer

$x\approx-0.693$

Work Step by Step

Using the properties of equality, the given equation, $ 7+3e^{-x}=13 ,$ is equivalent to \begin{array}{l}\require{cancel} 3e^{-x}=13-7 \\\\ 3e^{-x}=6 \\\\ e^{-x}=\dfrac{6}{3} \\\\ e^{-x}=2 .\end{array} Taking the natural logarithm of both sides and using the properties of logarithms, the value of the variable that satisfies the equation, $ e^{-x}=2 ,$ is \begin{array}{l}\require{cancel} \ln e^{-x}=\ln 2 \\\\ -x(\ln e)=\ln 2 \\\\ -x(1)=\ln 2 \\\\ -x=\ln 2 \\\\ x=-\ln 2 \\\\ x\approx-0.693 .\end{array}
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