Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![### Problem Statement
**Solve the equation:**
\[
4 \log_4 (x - 5) + \log_4 (x - 5) = 1
\]
### Explanation
This is a logarithmic equation where the base of the logarithm is 4. We need to find the value of \( x \) that satisfies the equation.
### Steps to Solve
1. **Combine the logarithmic terms:**
Since both terms have the same base and argument, they can be combined:
\[
(4 + 1) \log_4 (x - 5) = 1
\]
Simplifies to:
\[
5 \log_4 (x - 5) = 1
\]
2. **Simplify the equation:**
Divide both sides by 5:
\[
\log_4 (x - 5) = \frac{1}{5}
\]
3. **Convert the logarithmic equation to an exponential form:**
\[
x - 5 = 4^{\frac{1}{5}}
\]
4. **Solve for \( x \):**
\[
x = 4^{\frac{1}{5}} + 5
\]
5. **Final Solution:**
The solution is \( x = 4^{\frac{1}{5}} + 5 \).
This equation solving technique involves understanding both logarithmic properties and converting different forms of equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff73336ba-b7a2-41c3-9bb8-e711f85d25de%2F8515eb96-7b8e-4c93-abbf-2a94c6ff26f1%2Ft91x5m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Solve the equation:**
\[
4 \log_4 (x - 5) + \log_4 (x - 5) = 1
\]
### Explanation
This is a logarithmic equation where the base of the logarithm is 4. We need to find the value of \( x \) that satisfies the equation.
### Steps to Solve
1. **Combine the logarithmic terms:**
Since both terms have the same base and argument, they can be combined:
\[
(4 + 1) \log_4 (x - 5) = 1
\]
Simplifies to:
\[
5 \log_4 (x - 5) = 1
\]
2. **Simplify the equation:**
Divide both sides by 5:
\[
\log_4 (x - 5) = \frac{1}{5}
\]
3. **Convert the logarithmic equation to an exponential form:**
\[
x - 5 = 4^{\frac{1}{5}}
\]
4. **Solve for \( x \):**
\[
x = 4^{\frac{1}{5}} + 5
\]
5. **Final Solution:**
The solution is \( x = 4^{\frac{1}{5}} + 5 \).
This equation solving technique involves understanding both logarithmic properties and converting different forms of equations.
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