Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 7 - Section 7.4 - Systems of Nonlinear Equations in Two Variables - Exercise Set - Page 852: 79


The simplified solution set for the system of equations is $\left\{ \left( 8,2 \right) \right\}$.

Work Step by Step

We use the substitution method to solve the provided system of equations: $\left\{ \begin{align} & {{\log }_{y}}x=3....\text{(I)} \\ & {{\log }_{y}}\left( 4x \right)=5....\text{(II)} \end{align} \right.$ By using the logarithm rule ${{\log }_{m}}\left( ab \right)={{\log }_{m}}a+{{\log }_{m}}b $ in equation (II), we get $\begin{align} & {{\log }_{y}}\left( 4x \right)=5 \\ & {{\log }_{y}}x+{{\log }_{y}}4=5 \\ & {{\log }_{y}}x=5-{{\log }_{y}}4 \end{align}$ Substitute this value of ${{\log }_{y}}x $ in equation (I), $\begin{align} & {{\log }_{y}}x=3 \\ & 5-{{\log }_{y}}4=3 \\ & {{\log }_{y}}4=2 \end{align}$ The above equation is in a single variable. To solve this equation, transform the equation to the power of y on both sides. $\begin{align} & {{\log }_{y}}4=2 \\ & {{y}^{{{\log }_{y}}4}}={{y}^{2}} \end{align}$ We use the logarithm rule ${{m}^{{{\log }_{m}}a}}=a $ to simplify as shown below: $\begin{align} & {{y}^{{{\log }_{y}}4}}={{y}^{2}} \\ & 4={{y}^{2}} \\ & y=\pm 2 \end{align}$ The solution to this equation is $ y=2$ or $ y=-2$, but since the base of a logarithmic function cannot be a negative value, we ignore the solution $ y=-2$. Putting the values of y in equation (I) to find values for x, we get, For $ y=2$, $\begin{align} & {{\log }_{y}}x=3 \\ & {{\log }_{2}}x=3 \\ \end{align}$ To solve for $ x $ in this logarithmic equation, transform the equation to a power of 2 on both sides, $\begin{align} & {{\log }_{2}}x=3 \\ & {{2}^{{{\log }_{2}}x}}={{2}^{3}} \end{align}$ Apply the logarithm rule ${{m}^{{{\log }_{m}}a}}=a $ to simplify, $\begin{align} & {{2}^{{{\log }_{2}}x}}={{2}^{3}} \\ & x={{2}^{3}} \\ & x=8 \end{align}$ Hence, for $ y=2$, we have $ x=8$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.