College Algebra (10th Edition)

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

Chapter 6 - Section 6.5 - Properties of Logarithms - 6.5 Assess Your Understanding: 90

Answer

$y = \dfrac{Cx^2}{x+1}$`

Work Step by Step

RECALL: For positive real numbers M and N: (1) $\ln{M} = \ln{N} \longrightarrow M=N$ (2) Product Rule: $\ln{M} + \ln{N} = \ln{(MN)}$ (3) Quotient Rule: $\ln{M} - \ln{N} = \ln{\left(\dfrac{M}{N}\right)}$ (4) Power Rule: $r\cdot ln {M} = \ln{(M^r)}$ Use rule the Power Rule to obtain: $\ln{y}=2\ln{x}-\ln{(x+1)}+\ln{C} \\\ln{y} = \ln{x^2}-\ln{(x+1)}+\ln{C}$ Apply the Quotient Rule to the first two terms of the right side of the equation to obtain: $\ln{y} = \ln{\left(\dfrac{x^2}{x+1}\right)}+\ln{C}$ Apply the Product Rule to obtain: $\ln{y} = \ln{\left[\left(\dfrac{x^2}{x+1}\right) \cdot C\right]} \\\ln{y} = \ln{\left(\dfrac{Cx^2}{x+1}\right)}$ Use rule (1) above to obtain: $y = \dfrac{Cx^2}{x+1}$`
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