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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![Certainly! Here is the transcription suitable for an educational website:
---
**Mathematics Exercise**
1. **Solve the equation:**
\[ 4^x = 8^{x+1} \]
2. **Use the properties of logarithms to expand the logarithm:**
\[ \ln \left(\frac{5p^4}{\sqrt[3]{x}(m+2)^5}\right) \]
**Instructions:**
- For question 1, solve for \( x \) by rewriting the bases in terms of a common base if possible.
- For question 2, utilize the logarithm properties: product, quotient, and power rules to expand the given expression.
---
**Explanation of Concepts:**
- **Solving Exponential Equations:** Start by expressing both sides of the equation with the same base to simplify and solve for the variable.
- **Properties of Logarithms:**
- **Product Rule:** \(\ln(ab) = \ln a + \ln b\)
- **Quotient Rule:** \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\)
- **Power Rule:** \(\ln(a^b) = b \cdot \ln a\)
Use these properties to break down the logarithmic expression into simpler parts.
---
**Note**: The image includes a computer screen showing a mathematical problem set. There are no graphs or diagrams present.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee88b8a7-c213-48e1-bed2-7c496e1303f2%2Fb81ea4e2-3c1b-46c4-b371-4a16c3a232e2%2Fshblko9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcription suitable for an educational website:
---
**Mathematics Exercise**
1. **Solve the equation:**
\[ 4^x = 8^{x+1} \]
2. **Use the properties of logarithms to expand the logarithm:**
\[ \ln \left(\frac{5p^4}{\sqrt[3]{x}(m+2)^5}\right) \]
**Instructions:**
- For question 1, solve for \( x \) by rewriting the bases in terms of a common base if possible.
- For question 2, utilize the logarithm properties: product, quotient, and power rules to expand the given expression.
---
**Explanation of Concepts:**
- **Solving Exponential Equations:** Start by expressing both sides of the equation with the same base to simplify and solve for the variable.
- **Properties of Logarithms:**
- **Product Rule:** \(\ln(ab) = \ln a + \ln b\)
- **Quotient Rule:** \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\)
- **Power Rule:** \(\ln(a^b) = b \cdot \ln a\)
Use these properties to break down the logarithmic expression into simpler parts.
---
**Note**: The image includes a computer screen showing a mathematical problem set. There are no graphs or diagrams present.
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