Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 7 - Exponential and Logarithmic Functions - 7-6 Natural Logarithms - Practice and Problem-Solving Exercises - Page 481: 37

Answer

$x\approx2.197$

Work Step by Step

Subtract $7$ from each side to obtain: \begin{align*} 7-2e^{\frac{x}{2}}-7&=1-7\\\\ -2e^{\frac{x}{2}}&=-6\\\\ \frac{-2e^{\frac{x}{2}}}{-2}&=\frac{-6}{-2}\\\\ e^{\frac{x}{2}}&=3 \end{align*} RECALL: $$\ln{x}=y \longleftrightarrow e^y=x$$ Use the definition above to obtain: \begin{align*} \ln{3}&=\frac{x}{2}\\\\ 2(\ln3)&=2\left(\frac{x}{2}\right)\\\\ 2\ln3&=x \end{align*} Use a scientific calculator to obtain: $$x\approx 2.197$$
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