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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Transcribed Image Text:### Logarithm Question
Which of the following expressions is equivalent to \(\log\left(\frac{1}{2}\right)\)?
#### Options:
1. \(\frac{\log(1)}{\log(10)}\)
2. \(\ln(20)\)
3. \(\frac{\ln\left(\frac{1}{2}\right)}{\ln(10)}\)
4. \(\frac{\log(10)}{\log\left(\frac{1}{2}\right)}\)
#### Explanation
This question presents multiple choice options to identify the correct logarithmic expression equivalent to \(\log\left(\frac{1}{2}\right)\).
#### Additional Information
The expressions involve using both the natural logarithm (\(\ln\)) and the common logarithm (\(\log\)). Recognizing the relationships between these functions and their properties will help in selecting the correct answer.
For an educational approach:
- Review properties of logarithms including change of base formula.
- Understand how to express one logarithmic function in terms of another.
![### Change of Base Formula for Logarithms - Educational Guide
**Using the change of base formula, calculate \( \log_{\frac{1}{2}} 10 \):**
Here are the options given:
1. \( \frac{\log \left( \frac{1}{2} \right)}{\log (10)} \)
2. \(\ln (20)\)
3. \( \frac{\ln \left( \frac{1}{2} \right)}{\ln (10)} \)
4. \( \frac{\log (10)}{\log \left( \frac{1}{2} \right)} \)
In this problem, we employ the change of base formula to transform the given logarithm into a form that can be easily calculated using common logarithms (base 10) or natural logarithms (base \( e \)).
The general change of base formula is:
\[ \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \]
where \( c \) can be any positive number. Frequently, we use \( c = 10 \) or \( c = e \) because these logarithms can be easily computed with calculators.
Given the problem:
\[ \log_{\frac{1}{2}} 10 \]
Using the change of base formula with common logarithms \( (\log \) which typically represents log base 10):
\[ \log_{\frac{1}{2}} 10 = \frac{\log (10)}{\log \left( \frac{1}{2} \right)} \]
Thus, the correct answer is:
\[ \frac{\log (10)}{\log \left( \frac{1}{2} \right)} \]
This corresponds to the last option in the list.
**Explanation of Graphs or Diagrams:**
There are no graphs or diagrams in the image provided.
**Environmental Information:**
The screenshot indicates the following weather detail on the bottom left:
- **84°F**
- **Partly sunny**
This information is likely from a desktop's taskbar/weather widget and is additional context captured within the screenshot.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda090e0a-52ff-407b-b6d7-bd2e820f5422%2Fd154a4c7-b4d6-4f41-bf4e-9f6e0b2e9b2a%2Fdpfxxue_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Change of Base Formula for Logarithms - Educational Guide
**Using the change of base formula, calculate \( \log_{\frac{1}{2}} 10 \):**
Here are the options given:
1. \( \frac{\log \left( \frac{1}{2} \right)}{\log (10)} \)
2. \(\ln (20)\)
3. \( \frac{\ln \left( \frac{1}{2} \right)}{\ln (10)} \)
4. \( \frac{\log (10)}{\log \left( \frac{1}{2} \right)} \)
In this problem, we employ the change of base formula to transform the given logarithm into a form that can be easily calculated using common logarithms (base 10) or natural logarithms (base \( e \)).
The general change of base formula is:
\[ \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \]
where \( c \) can be any positive number. Frequently, we use \( c = 10 \) or \( c = e \) because these logarithms can be easily computed with calculators.
Given the problem:
\[ \log_{\frac{1}{2}} 10 \]
Using the change of base formula with common logarithms \( (\log \) which typically represents log base 10):
\[ \log_{\frac{1}{2}} 10 = \frac{\log (10)}{\log \left( \frac{1}{2} \right)} \]
Thus, the correct answer is:
\[ \frac{\log (10)}{\log \left( \frac{1}{2} \right)} \]
This corresponds to the last option in the list.
**Explanation of Graphs or Diagrams:**
There are no graphs or diagrams in the image provided.
**Environmental Information:**
The screenshot indicates the following weather detail on the bottom left:
- **84°F**
- **Partly sunny**
This information is likely from a desktop's taskbar/weather widget and is additional context captured within the screenshot.
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