/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 79 Solve the systems. $$\left\\{\... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 79

Solve the systems. $$\left\\{\begin{array}{l} \log _{y} x=3 \\ \log _{y}(4 x)=5 \end{array}\right.$$

Short Answer

Expert verified
The solution of the system is (x, y) = (8, 2).

Step by step solution

01

Conversion to Exponential Form

The first step is to convert these logarithmic equations into their exponential forms. The relationship between logarithmic and exponential forms is: if \( \log_{b} a = c \), then \( b^{c} = a \). Using this relationship, the first equation \( \log_{y} x = 3 \) becomes \( y^{3} = x \), and the second equation \( \log_{y}(4 x) = 5 \) becomes \( y^{5} = 4x \).
02

Substitution

Next, we substitute the value of \( x \) from the first equation into the second equation. That is, we substitute \( y^{3} \) for \( x \) in the equation \( y^{5} = 4x \) to obtain \( y^{5} = 4y^{3} \).
03

Solve for y

In the equation \( y^{5} = 4y^{3} \), we can solve for \( y \) by dividing both sides by \( y^{3} \) (assuming \( y \neq 0 \)), which gives us \( y^{2} = 4 \). The solution for \( y \) is then found by taking the square root of both sides, thus \( y = \sqrt{4} = \pm 2 \). We need to remember that for the logarithmic forms, \( y \) must be positive
04

Solve for x

We then substitute \( y = 2\) into \( y^{3} = x \) to solve for \( x \) to get \( x = 2^{3} = 8 \). Verify this as a solution: it should satisfy both original equations. \( \log_{2} 8 = 3 \) and \( \log_{2}(4 \times 8) = 5 \), therefore (x, y) = (8, 2) is the solution to the system of equations.

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Most popular questions from this chapter

This will help you prepare for the material covered in the first section of the next chapter. Evaluate \(\frac{(-1)^{n}}{3^{n}-1}\) for \(n=1,2,3,\) and 4

(GRAPH CANT COPY) Find the coordinates of the vertex for the horizontal parabola defined by the given equation. $$x=-2(y+5)^{2}-1$$

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. $$x-7-8 y=y^{2}$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{l}x=(y+2)^{2}-1 \\\ (x-2)^{2}+(y+2)^{2}=1\end{array}\right.$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=\frac{1}{2}(y+2)^{2}+1$$
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