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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![**Topic: Simplifying Logarithmic Expressions**
**Objective:**
Rewrite the following expression as a single logarithm:
\[ 3(\log_3 x + 2\log_3 y - 4\log_3 z) \]
**Instructions:**
To simplify this expression, apply the following logarithmic properties:
1. **Product Property**: \(\log_b (MN) = \log_b M + \log_b N\)
2. **Quotient Property**: \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\)
3. **Power Property**: \(a \log_b M = \log_b (M^a)\)
**Steps:**
1. Apply the power property to each term inside the parenthesis:
- \(\log_3 x\) remains \(\log_3 x\)
- Convert \(2\log_3 y\) to \(\log_3 (y^2)\)
- Convert \(-4\log_3 z\) to \(-\log_3 (z^4)\)
2. Combine the logarithms inside the parenthesis:
- Use the product and quotient properties:
- \(\log_3 x + \log_3(y^2) - \log_3(z^4) = \log_3 \left(\frac{x \cdot y^2}{z^4}\right)\)
3. Apply the multiplied factor outside the parenthesis using the power property:
- \(3 \cdot \log_3 \left(\frac{x \cdot y^2}{z^4}\right) = \log_3 \left(\left(\frac{x \cdot y^2}{z^4}\right)^3\right)\)
4. Simplified expression:
- \(\log_3 \left(\frac{x^3 \cdot y^6}{z^{12}}\right)\)
By following these steps, the given expression is rewritten as a single logarithm:
\[ \log_3 \left(\frac{x^3 \cdot y^6}{z^{12}}\right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f805aea-efd1-4603-9b0d-77a77b2a0177%2Fffa38286-46ed-4fcf-9043-1baf6954d5ae%2Fecq9ey_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Simplifying Logarithmic Expressions**
**Objective:**
Rewrite the following expression as a single logarithm:
\[ 3(\log_3 x + 2\log_3 y - 4\log_3 z) \]
**Instructions:**
To simplify this expression, apply the following logarithmic properties:
1. **Product Property**: \(\log_b (MN) = \log_b M + \log_b N\)
2. **Quotient Property**: \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\)
3. **Power Property**: \(a \log_b M = \log_b (M^a)\)
**Steps:**
1. Apply the power property to each term inside the parenthesis:
- \(\log_3 x\) remains \(\log_3 x\)
- Convert \(2\log_3 y\) to \(\log_3 (y^2)\)
- Convert \(-4\log_3 z\) to \(-\log_3 (z^4)\)
2. Combine the logarithms inside the parenthesis:
- Use the product and quotient properties:
- \(\log_3 x + \log_3(y^2) - \log_3(z^4) = \log_3 \left(\frac{x \cdot y^2}{z^4}\right)\)
3. Apply the multiplied factor outside the parenthesis using the power property:
- \(3 \cdot \log_3 \left(\frac{x \cdot y^2}{z^4}\right) = \log_3 \left(\left(\frac{x \cdot y^2}{z^4}\right)^3\right)\)
4. Simplified expression:
- \(\log_3 \left(\frac{x^3 \cdot y^6}{z^{12}}\right)\)
By following these steps, the given expression is rewritten as a single logarithm:
\[ \log_3 \left(\frac{x^3 \cdot y^6}{z^{12}}\right) \]
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