Chapter 27: Problem 48
The solution set of \(|x+2|^{\log _{10}\left(x^{2}+6 x+9\right)}=1\) is (1) \(\\{-3,-4\\}\) (2) \(\\{0,-3\\}\) (3) \(\\{-4,-1\\}\) (4) \(\\{-3,-1\\}\)
Short Answer
Expert verified
Answer: (1) \(\{-3, -4\}\)
Step by step solution
01
Rewrite the given equation
The equation we need to solve is:
\(|x+2|^{\log _{10}\left(x^{2}+6 x+9\right)}=1\)
02
Properties of exponents
We know some properties of exponents that will help us to simplify the equation:
1. Any value to the power of 0 is 1: \(a^{0} = 1\)
2. Any number to the power of 1 is itself: \(a^{1} = a\)
Therefore, if the given equation is equal to 1, it must mean that the exponent, which is the logarithm, is equal to 0 or the base, which is the absolute value of (x+2), is equal to 1.
03
Solve for x when the exponent is 0
When the exponent is equal to 0, the logarithm must be equal to 0 as well:
\(\log _{10}\left(x^{2}+6 x+9\right) = 0\)
We need to solve for x:
\(x^{2}+6 x+9 = 10^{0}\)
\(x^{2}+6 x+9 = 1\)
\(x^{2}+6 x+8 = 0\)
Now factor the quadratic equation:
\((x+2)(x+4)=0\)
Which gives us two solutions:
1. x = -2
2. x = -4
04
Solve for x when the base is 1
When the base is equal to 1, the absolute value of (x+2) must be equal to 1 as well:
\(|x+2| = 1\)
This can be written as two separate equations:
1. \(x+2 = 1\), which gives \(x = -1\)
2. \(-(x+2) = 1\), which gives \(x = -3\)
05
Identify the correct answer set
The solution set for x consists of the numbers {-2, -4, -1, -3}. Among the given answer sets, only one contains these values:
(1) \(\{-3, -4\}\)
The correct answer is thus option (1).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Properties of Exponents
The properties of exponents are fundamental rules in mathematics that help simplify expressions involving powers. They provide a foundation for understanding how numbers and variables with exponents behave.
One key property is that any number raised to the power of zero is always 1. For example, for any non-zero number \(a\), we have \(a^0 = 1\). This property is useful because it allows us to simplify expressions where the exponent equals zero.
Another essential property is that any number raised to the power of one is the number itself. That is, \(a^1 = a\). This makes it clear that multiplying by one does not change the base value.
Understanding these properties allows us to solve equations where the base raised to a power is set equal to 1, like in this exercise. You need to determine if it's because the exponent is zero or the base itself is 1. These two cases will guide to different solutions for equations, a critical skill in algebra.
One key property is that any number raised to the power of zero is always 1. For example, for any non-zero number \(a\), we have \(a^0 = 1\). This property is useful because it allows us to simplify expressions where the exponent equals zero.
Another essential property is that any number raised to the power of one is the number itself. That is, \(a^1 = a\). This makes it clear that multiplying by one does not change the base value.
Understanding these properties allows us to solve equations where the base raised to a power is set equal to 1, like in this exercise. You need to determine if it's because the exponent is zero or the base itself is 1. These two cases will guide to different solutions for equations, a critical skill in algebra.
Logarithmic Functions
Logarithmic functions are inverse operations of exponential functions. They help us understand the relationship between numbers in terms of powers and roots.
The logarithmic function with base 10, often written as \( \log_{10} \), is used to find which power 10 must be raised to yield a certain number. For example, \( \log_{10}(100) = 2 \) because \( 10^2 = 100 \).
In solving equations where the logarithm equals zero, such as \( \log_{10}(x^2 + 6x + 9) = 0 \), we interpret this to mean that the expression inside the log function equals 1. This is because \( \log_{10}(1) = 0 \) since any base raised to the zero power is one.
This property allows us to rewrite equations and find solutions. Often, it informs us about the values that the variable inside the logarithm can have. By solving these, you can identify possible solutions to your main equation.
The logarithmic function with base 10, often written as \( \log_{10} \), is used to find which power 10 must be raised to yield a certain number. For example, \( \log_{10}(100) = 2 \) because \( 10^2 = 100 \).
In solving equations where the logarithm equals zero, such as \( \log_{10}(x^2 + 6x + 9) = 0 \), we interpret this to mean that the expression inside the log function equals 1. This is because \( \log_{10}(1) = 0 \) since any base raised to the zero power is one.
This property allows us to rewrite equations and find solutions. Often, it informs us about the values that the variable inside the logarithm can have. By solving these, you can identify possible solutions to your main equation.
Absolute Value Equations
Absolute value equations are equations where the variable is within an absolute value expression, such as \(|x+2|\). The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative number.
For example, the absolute value of both -3 and 3 is 3, denoted as \(|-3| = 3\) and \(|3| = 3\). In equations, this means that if \(|x+2| = 1\), two possible scenarios exist: either \(x+2 = 1\) or \(-(x+2) = 1\). Each scenario provides a different solution for \(x\).
This dual possibility is crucial in solving absolute value equations as it can lead to multiple solutions. So, always remember to consider both the positive and negative scenarios for the expression within the absolute value. Understanding absolute value equations allows us to delve deeper into solution sets and find all possible values that satisfy the original equation.
For example, the absolute value of both -3 and 3 is 3, denoted as \(|-3| = 3\) and \(|3| = 3\). In equations, this means that if \(|x+2| = 1\), two possible scenarios exist: either \(x+2 = 1\) or \(-(x+2) = 1\). Each scenario provides a different solution for \(x\).
This dual possibility is crucial in solving absolute value equations as it can lead to multiple solutions. So, always remember to consider both the positive and negative scenarios for the expression within the absolute value. Understanding absolute value equations allows us to delve deeper into solution sets and find all possible values that satisfy the original equation.
