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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 54 (page 773)

In Problems $25 - 54$ , solve each system. Use any method you wish.
$\{\begin{cases} \ln x = 5 \ln y \\ \log_{2} x = 3 + 2 \log_{2} y \end{cases}$

Short Answer

Expert verified

The solution for $\{\begin{cases} \ln x = 5 \ln y \\ \log_{2} x = 3 + 2 \log_{2} y \end{cases}$ is, $(32, 2)$ .

Step by step solution

01

Step 1 Given that,

$\{\begin{cases} \ln x = 5 \ln y \\ \log_{2} x = 3 + 2 \log_{2} y \end{cases}$

Consider the first equation and simplify it using the logarithmic properties.

$\ln x = 5 \ln y \ln x - 5 \ln y = 0 \ln x - \ln y^{5} = 0 \ln (\frac{x}{y^{5}}) = 0 \frac{x}{y^{5}} = e^{0} \frac{x}{y^{5}} = 1 x = y^{5}$

02

Step 2 Now consider the second equation and simplify it by using the logarithmic properties.

$\log_{2} x = 3 + 2 \log_{2} y \log_{2} x = 3 + \log_{2} y^{2} \log_{2} x - \log_{2} y^{2} = 3 \log_{2} (\frac{x}{y^{2}}) = 3 \frac{x}{y^{2}} = {(2)}^{3} x = 8 y^{2}$

03

Step 3 Now equate x=y5 and x=8y2.

$y^{5} = 8 y^{2} \frac{y^{5}}{y^{2}} = \frac{8 y^{2}}{y^{2}} y^{3} = 8 y = 2$

Substitute $y = 2$ in $x = y^{5}$ .

$x = {(2)}^{5} x = 32$

The solution for the given system of equation is, $(32, 2)$ .

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Most popular questions from this chapter

In Problems 35鈥�42, graph each system of inequalities by hand.

$\{\begin{cases} x^{2} + y^{2} 鈮� 9 \\ x + y 鈮� 3 \end{cases}$

Graph each inequality by hand.

$3 x + 2 y 鈮� 6$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$2 x + y - 3 z = 0 - 2 x + 2 y + z = - 7 3 x - 4 y - 3 z = 7$

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$y = \sqrt{x}; y = 2 - x$

solve each system of equations. If the system has no solution, say that it is inconsistent.

$2 x + y = 4 - 2 y + 4 z = 0 3 x - 2 z = - 11$

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