College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter5: Exponential And Logarithmic Functions
Section5.3: Logarithmic Functions And Their Graphs
Problem 140E
Related questions
Question
![### Problem:
**Question 94:**
Express \( y \) as a function of \( x \). The constant \( C \) is a positive number.
\[ \ln(y + 4) = 5x + \ln C \]
### Explanation:
In this problem, we are given a logarithmic equation and need to express \( y \) as a function of \( x \).
### Solution:
1. **Given Equation:**
\[ \ln(y + 4) = 5x + \ln C \]
2. **Use Properties of Logarithms:**
Isolate the logarithmic expression on one side.
\[ \ln(y + 4) = \ln(Ce^{5x}) \]
This step uses the property of logarithms: \( \ln(a) + \ln(b) = \ln(ab) \).
3. **Exponentiate both sides to remove the natural log:**
\[ y + 4 = Ce^{5x} \]
4. **Solve for \( y \):**
\[ y = Ce^{5x} - 4 \]
### Result:
The function \( y \) in terms of \( x \) is:
\[ y = Ce^{5x} - 4 \]
Where \( C \) is a positive constant.
This completes the transcription and explanation for expressing \( y \) as a function of \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F349eaa27-36eb-4309-87e3-9ff3181a3e80%2Ff1482c7e-937d-4e33-ab1c-8d7195aac23d%2F2skz3t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem:
**Question 94:**
Express \( y \) as a function of \( x \). The constant \( C \) is a positive number.
\[ \ln(y + 4) = 5x + \ln C \]
### Explanation:
In this problem, we are given a logarithmic equation and need to express \( y \) as a function of \( x \).
### Solution:
1. **Given Equation:**
\[ \ln(y + 4) = 5x + \ln C \]
2. **Use Properties of Logarithms:**
Isolate the logarithmic expression on one side.
\[ \ln(y + 4) = \ln(Ce^{5x}) \]
This step uses the property of logarithms: \( \ln(a) + \ln(b) = \ln(ab) \).
3. **Exponentiate both sides to remove the natural log:**
\[ y + 4 = Ce^{5x} \]
4. **Solve for \( y \):**
\[ y = Ce^{5x} - 4 \]
### Result:
The function \( y \) in terms of \( x \) is:
\[ y = Ce^{5x} - 4 \]
Where \( C \) is a positive constant.
This completes the transcription and explanation for expressing \( y \) as a function of \( x \).
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