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
![**Condensing Logarithms**
1. **Rewrite the following as a condensed, single logarithm:**
\( 5\log_2(x) - 2\log_2(y) + \log_2(w) \)
### Explanation
To condense the given expression into a single logarithm, apply the properties of logarithms: the power rule, the product rule, and the quotient rule.
- **Power Rule:** \( a\log_b(m) = \log_b(m^a) \)
- **Product Rule:** \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \)
- **Quotient Rule:** \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \)
Steps:
1. Apply the power rule:
- \( 5\log_2(x) = \log_2(x^5) \)
- \( -2\log_2(y) = \log_2(y^{-2}) \)
2. Combine using the product and quotient rules:
\[
\log_2(x^5) + \log_2(w) - \log_2(y^2) = \log_2\left(\frac{x^5 \cdot w}{y^2}\right)
\]
Thus, the expression \(\log_2\left(\frac{x^5 \cdot w}{y^2}\right)\) is the condensed, single logarithm form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcbb2b57d-b82b-4c44-a3ed-52ccdcaab288%2F5b9059d6-7e21-48d0-b676-cec1b609e171%2F2ut6v6mh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Condensing Logarithms**
1. **Rewrite the following as a condensed, single logarithm:**
\( 5\log_2(x) - 2\log_2(y) + \log_2(w) \)
### Explanation
To condense the given expression into a single logarithm, apply the properties of logarithms: the power rule, the product rule, and the quotient rule.
- **Power Rule:** \( a\log_b(m) = \log_b(m^a) \)
- **Product Rule:** \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \)
- **Quotient Rule:** \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \)
Steps:
1. Apply the power rule:
- \( 5\log_2(x) = \log_2(x^5) \)
- \( -2\log_2(y) = \log_2(y^{-2}) \)
2. Combine using the product and quotient rules:
\[
\log_2(x^5) + \log_2(w) - \log_2(y^2) = \log_2\left(\frac{x^5 \cdot w}{y^2}\right)
\]
Thus, the expression \(\log_2\left(\frac{x^5 \cdot w}{y^2}\right)\) is the condensed, single logarithm form.
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