College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 6 - Section 6.4 - Logarithmic Functions - 6.4 Assess Your Understanding: 109

Answer

$x = 5\ln{\frac{7}{5}}$

Work Step by Step

Divide both sides of the equation by 5 to obtain: $\dfrac{5e^{0.2x}}{5} = \dfrac{7}{5} \\e^{0.2x}=\dfrac{7}{5}$ Take the natural logarithm of both sides to obtain: $\ln{e^{0.2x}} = \ln{\frac{7}{5}}$ Note that $\ln{e^x} = x$. Thus, the equation above is equivalent to: $0.2x = \ln{\frac{7}{5}}$ Divide by $0.2$ on both sides of the equation to obtain: $\dfrac{0.2x}{2} = \dfrac{\ln{\frac{7}{5}}}{0.2} \\x = \dfrac{\ln{1.4}}{0.2}$ Note that $0.2=\frac{1}{5}$. This means that the equation above is equivalent to: $x=\dfrac{\ln{\frac{7}{5}}}{\frac{1}{5}} \\x = (\ln{\frac{7}{5}}) \cdot \dfrac{5}{1} \\x = 5\ln{\frac{7}{5}}$
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