Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Question 1: Logarithmic Expression Simplification**
**Task**: Rewrite \(2\ln(x) - 3\ln(y) + \ln(z)\) as a single logarithm.
**Solution Approach**:
To combine multiple logarithms into a single logarithmic expression, we can use the properties of logarithms:
1. **Power Rule**: \( a\ln(b) = \ln(b^a) \)
2. **Product Rule**: \( \ln(a) + \ln(b) = \ln(a \times b) \)
3. **Quotient Rule**: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \)
**Steps**:
1. Apply the Power Rule:
- Transform \(2\ln(x)\) to \(\ln(x^2)\)
- Transform \(-3\ln(y)\) to \(\ln(y^{-3})\)
2. Substitute back into the expression:
- \(\ln(x^2) + \ln(z) - \ln(y^3)\)
3. Use the Product Rule:
- Combine \(\ln(x^2)\) and \(\ln(z)\) to get \(\ln(x^2z)\)
4. Apply the Quotient Rule:
- Subtract \(\ln(y^3)\) from \(\ln(x^2z)\) to get \(\ln\left(\frac{x^2z}{y^3}\right)\)
**Final Expression**:
The expression \(2\ln(x) - 3\ln(y) + \ln(z)\) can be rewritten as a single logarithm:
\[ \ln\left(\frac{x^2z}{y^3}\right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F322a2375-fbba-45b8-8629-b5912bbd9c12%2Ff266959d-1d9b-4731-b345-14e9ba5a912e%2Fooye8ap_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 1: Logarithmic Expression Simplification**
**Task**: Rewrite \(2\ln(x) - 3\ln(y) + \ln(z)\) as a single logarithm.
**Solution Approach**:
To combine multiple logarithms into a single logarithmic expression, we can use the properties of logarithms:
1. **Power Rule**: \( a\ln(b) = \ln(b^a) \)
2. **Product Rule**: \( \ln(a) + \ln(b) = \ln(a \times b) \)
3. **Quotient Rule**: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \)
**Steps**:
1. Apply the Power Rule:
- Transform \(2\ln(x)\) to \(\ln(x^2)\)
- Transform \(-3\ln(y)\) to \(\ln(y^{-3})\)
2. Substitute back into the expression:
- \(\ln(x^2) + \ln(z) - \ln(y^3)\)
3. Use the Product Rule:
- Combine \(\ln(x^2)\) and \(\ln(z)\) to get \(\ln(x^2z)\)
4. Apply the Quotient Rule:
- Subtract \(\ln(y^3)\) from \(\ln(x^2z)\) to get \(\ln\left(\frac{x^2z}{y^3}\right)\)
**Final Expression**:
The expression \(2\ln(x) - 3\ln(y) + \ln(z)\) can be rewritten as a single logarithm:
\[ \ln\left(\frac{x^2z}{y^3}\right) \]
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