Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Using Properties of Logarithms to Expand**
**Problem 24)**
Expand the logarithmic expression using the properties of logarithms:
\[ \log_2 \left( \frac{x^2 (x - 9)}{y} \right) \]
### Expansion Steps
1. **Apply the Division Property**:
The logarithm of a quotient is the difference of the logarithms.
\[ \log_2 \left( \frac{x^2 (x - 9)}{y} \right) = \log_2 (x^2 (x - 9)) - \log_2 (y) \]
2. **Apply the Multiplication Property**:
The logarithm of a product is the sum of the logarithms.
\[ \log_2 (x^2 (x - 9)) = \log_2 (x^2) + \log_2 (x - 9) \]
3. **Apply the Power Property**:
The logarithm of a power can be expressed as the exponent times the logarithm of the base.
\[ \log_2 (x^2) = 2 \log_2 (x) \]
Combining all parts together:
\[ \log_2 \left( \frac{x^2 (x - 9)}{y} \right) = 2 \log_2 (x) + \log_2 (x - 9) - \log_2 (y) \]
### Final Expanded Form
\[ \log_2 \left( \frac{x^2 (x - 9)}{y} \right) = 2 \log_2 (x) + \log_2 (x - 9) - \log_2 (y) \]
### Explanation of Graphs or Diagrams
The image provided did not contain any graphs or diagrams directly associated with the problem. The portion of the image shown focuses on the mathematics problem which asks to use the properties of logarithms to expand a given logarithmic expression. The properties used include the division property, multiplication property, and power property, with the final answer combining these expansions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9da2a76a-87f7-4e93-94c9-a00561324ff3%2F4ca89f7a-27a6-484d-8908-7eb97154aad8%2Fc6la47b_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Properties of Logarithms to Expand**
**Problem 24)**
Expand the logarithmic expression using the properties of logarithms:
\[ \log_2 \left( \frac{x^2 (x - 9)}{y} \right) \]
### Expansion Steps
1. **Apply the Division Property**:
The logarithm of a quotient is the difference of the logarithms.
\[ \log_2 \left( \frac{x^2 (x - 9)}{y} \right) = \log_2 (x^2 (x - 9)) - \log_2 (y) \]
2. **Apply the Multiplication Property**:
The logarithm of a product is the sum of the logarithms.
\[ \log_2 (x^2 (x - 9)) = \log_2 (x^2) + \log_2 (x - 9) \]
3. **Apply the Power Property**:
The logarithm of a power can be expressed as the exponent times the logarithm of the base.
\[ \log_2 (x^2) = 2 \log_2 (x) \]
Combining all parts together:
\[ \log_2 \left( \frac{x^2 (x - 9)}{y} \right) = 2 \log_2 (x) + \log_2 (x - 9) - \log_2 (y) \]
### Final Expanded Form
\[ \log_2 \left( \frac{x^2 (x - 9)}{y} \right) = 2 \log_2 (x) + \log_2 (x - 9) - \log_2 (y) \]
### Explanation of Graphs or Diagrams
The image provided did not contain any graphs or diagrams directly associated with the problem. The portion of the image shown focuses on the mathematics problem which asks to use the properties of logarithms to expand a given logarithmic expression. The properties used include the division property, multiplication property, and power property, with the final answer combining these expansions.
![### Algebra and Logarithms: Solving Logarithmic Equations
#### Example Problem: Solving a Logarithmic Equation
Consider the following logarithmic equation:
\[ \ln x + \ln (x - 1) = \ln 72 \]
We aim to find the value of \(x\) that satisfies this equation.
#### Step-by-Step Solution
1. **Combine the Logarithmic Expressions**:
Using the property of logarithms, \(\ln a + \ln b = \ln (a \cdot b)\), we can combine the logarithms on the left-hand side of the equation:
\[ \ln [x \cdot (x - 1)] = \ln 72 \]
2. **Simplify the Expression Inside the Logarithm**:
This simplifies to:
\[ \ln (x^2 - x) = \ln 72 \]
3. **Remove the Logarithms**:
Since the logarithms on both sides are equal, we can set the arguments inside the logarithms equal to each other:
\[ x^2 - x = 72 \]
4. **Form a Quadratic Equation**:
Rearrange the equation to form a standard quadratic equation:
\[ x^2 - x - 72 = 0 \]
5. **Solve the Quadratic Equation**:
Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -1\), and \(c = -72\):
\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1}
\]
\[
x = \frac{1 \pm \sqrt{1 + 288}}{2}
\]
\[
x = \frac{1 \pm \sqrt{289}}{2}
\]
\[
x = \frac{1 \pm 17}{2}
\]
This gives us two potential solutions:
\[
x = \frac{1 + 17}{2} = 9 \quad \text{and} \quad x = \frac{1 - 17](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9da2a76a-87f7-4e93-94c9-a00561324ff3%2F4ca89f7a-27a6-484d-8908-7eb97154aad8%2Fu4cnkpx_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Algebra and Logarithms: Solving Logarithmic Equations
#### Example Problem: Solving a Logarithmic Equation
Consider the following logarithmic equation:
\[ \ln x + \ln (x - 1) = \ln 72 \]
We aim to find the value of \(x\) that satisfies this equation.
#### Step-by-Step Solution
1. **Combine the Logarithmic Expressions**:
Using the property of logarithms, \(\ln a + \ln b = \ln (a \cdot b)\), we can combine the logarithms on the left-hand side of the equation:
\[ \ln [x \cdot (x - 1)] = \ln 72 \]
2. **Simplify the Expression Inside the Logarithm**:
This simplifies to:
\[ \ln (x^2 - x) = \ln 72 \]
3. **Remove the Logarithms**:
Since the logarithms on both sides are equal, we can set the arguments inside the logarithms equal to each other:
\[ x^2 - x = 72 \]
4. **Form a Quadratic Equation**:
Rearrange the equation to form a standard quadratic equation:
\[ x^2 - x - 72 = 0 \]
5. **Solve the Quadratic Equation**:
Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -1\), and \(c = -72\):
\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1}
\]
\[
x = \frac{1 \pm \sqrt{1 + 288}}{2}
\]
\[
x = \frac{1 \pm \sqrt{289}}{2}
\]
\[
x = \frac{1 \pm 17}{2}
\]
This gives us two potential solutions:
\[
x = \frac{1 + 17}{2} = 9 \quad \text{and} \quad x = \frac{1 - 17
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