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

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.

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

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

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

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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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

Step by step solution

Step 1 Given that,

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

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

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In Problems 25-54, solve each system. Use any method you wish.lnx=5lnylog2x=3+2log2y

Short Answer

Step by step solution

Step 1 Given that,

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

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Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

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