Solve the systems in Exercises 79–80.{log x2=y+3log x=y−1\left\{\begin{array}{l}\text{log }$$x^2=y+3$$\\ \text{log }x^{}=y-1\end{array}\right.

Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 80

Chapter 6, Problem 80

Solve the systems in Exercises 79–80.
${\begin{matrix} log x^{2} = y + 3 \\ log x^{} = y - 1 \end{matrix}$

Verified step by step guidance

Rewrite the given system of equations for clarity:

\log x^{2} = y + 3

and

\log x = y - 1

Recall the logarithm property:

\log x^{2} = 2 \log x

. Use this to rewrite the first equation as

2 \log x = y + 3

Substitute

\log x

from the second equation into the first. Since

\log x = y - 1

, replace

\log x

in the first equation with

y - 1

to get

2(y - 1) = y + 3

Solve the resulting linear equation for

y

: expand and simplify

2y - 2 = y + 3

, then isolate

y

Once you find

y

, substitute it back into

\log x = y - 1

to find

\log x

, and then solve for

x

by rewriting the logarithmic equation in exponential form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Understanding the properties of logarithms, such as the power rule (log a^b = b log a), is essential for manipulating and simplifying logarithmic expressions. This allows rewriting terms like log x^2 as 2 log x, facilitating easier comparison and solving of equations.

Solving Systems of Equations

Solving systems of equations involves finding values for variables that satisfy all given equations simultaneously. Techniques include substitution and elimination, which help reduce the system to a single-variable equation for easier solution.

Relationship Between Logarithmic and Linear Expressions

Recognizing how logarithmic expressions relate to linear equations is crucial. For example, expressing log x in terms of y allows converting the system into linear form, making it easier to solve using algebraic methods.

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Logarithms Introduction

Related Practice

Textbook Question

Solve: x⁴+2x³−x²−4x−2=0

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Textbook Question

Exercises 86–88 will help you prepare for the material covered in the first section of the next chapter. a. Does (4, −1) satisfy x + 2y = 2? b. Does (4, -1) satisfy x- 2y= 6?

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Textbook Question

Use a system of linear equations to solve Exercises 73–84. How many ounces of a 50% alcohol solution must be mixed with 80 ounces of a 20% alcohol solution to make a 40% alcohol solution?

Textbook Question

Use a system of linear equations to solve Exercises 73–84. How many ounces of a 15% alcohol solution must be mixed with 4 ounces of a 20% alcohol solution to make a 17% alcohol solution?

Textbook Question

Solve the systems in Exercises 79–80.

${\begin{matrix} \log_{y} x=3 \\ \log_{y} (4x)=5 \end{matrix}$

Solve the systems in Exercises 79–80.{log x2=y+3log x=y−1\(\left\)\{\(\begin{array}{l}\[\text{log }\)x^2=y+3\\ \(\text{log }\)x^{}=y-1\(\end{array}\]\right\).

Verified video answer for a similar problem:

Key Concepts

Properties of Logarithms

Solving Systems of Equations

Relationship Between Logarithmic and Linear Expressions

Solve the systems in Exercises 79–80.
${\begin{matrix} log x^{2} = y + 3 \\ log x^{} = y - 1 \end{matrix}$