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,...
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![**Logarithmic Equation Solving Example**
*Objective: Solve for \( x \) in the equation \( 6^{x-4} = 7 \) using the change of base formula.*
### Problem Statement:
Solve \( 6^{x-4} = 7 \) for \( x \) using the change of base formula:
\[ \log_b y = \frac{\log y}{\log b} \]
### Answer Choices:
- \( \) 5.209
- \( \) 4.827
- \( \) -2.791
- \( \) -3.173
### Solution Steps:
1. Start with the equation:\[ 6^{x-4} = 7 \]
2. Apply the natural logarithm (or common logarithm) to both sides:
\[ \log (6^{x-4}) = \log 7 \]
3. Use the logarithm power rule (\( \log a^b = b \log a \)):
\[ (x-4) \log 6 = \log 7 \]
4. Solve for \( x-4 \) by isolating the variable on one side:
\[ x-4 = \frac{\log 7}{\log 6} \]
5. Finally, solve for \( x \):
\[ x = 4 + \frac{\log 7}{\log 6} \]
### Verify the Answer
Now substitute each answer choice into the solution to find the correct value:
1. **\( x = 5.209 \)**:
\[ 5.209 - 4 = 1.209 \]
\[ 6^{1.209} \approx 7 \] (Approximately true, calculated value close to 7)
2. **\( x = 4.827 \)**:
\[ 4.827 - 4 = 0.827 \]
\[ 6^{0.827} \not\approx 7 \] (Approximately not true, calculated value not close to 7)
3. **\( x = -2.791 \)**:
\[ -2.791 - 4 = -6.791 \]
\[ 6^{-6.791} \not\approx 7 \] (Approximately not true, calculated value not close to 7)
4. **\( x = -3.173 \)**:
\[ -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F355372c0-0b06-4568-8d9a-cddd4364f7ea%2Fb9df24cc-5a35-456a-aec4-1711c1654b3f%2Fisbbvyr_processed.png&w=3840&q=75)
Transcribed Image Text:**Logarithmic Equation Solving Example**
*Objective: Solve for \( x \) in the equation \( 6^{x-4} = 7 \) using the change of base formula.*
### Problem Statement:
Solve \( 6^{x-4} = 7 \) for \( x \) using the change of base formula:
\[ \log_b y = \frac{\log y}{\log b} \]
### Answer Choices:
- \( \) 5.209
- \( \) 4.827
- \( \) -2.791
- \( \) -3.173
### Solution Steps:
1. Start with the equation:\[ 6^{x-4} = 7 \]
2. Apply the natural logarithm (or common logarithm) to both sides:
\[ \log (6^{x-4}) = \log 7 \]
3. Use the logarithm power rule (\( \log a^b = b \log a \)):
\[ (x-4) \log 6 = \log 7 \]
4. Solve for \( x-4 \) by isolating the variable on one side:
\[ x-4 = \frac{\log 7}{\log 6} \]
5. Finally, solve for \( x \):
\[ x = 4 + \frac{\log 7}{\log 6} \]
### Verify the Answer
Now substitute each answer choice into the solution to find the correct value:
1. **\( x = 5.209 \)**:
\[ 5.209 - 4 = 1.209 \]
\[ 6^{1.209} \approx 7 \] (Approximately true, calculated value close to 7)
2. **\( x = 4.827 \)**:
\[ 4.827 - 4 = 0.827 \]
\[ 6^{0.827} \not\approx 7 \] (Approximately not true, calculated value not close to 7)
3. **\( x = -2.791 \)**:
\[ -2.791 - 4 = -6.791 \]
\[ 6^{-6.791} \not\approx 7 \] (Approximately not true, calculated value not close to 7)
4. **\( x = -3.173 \)**:
\[ -
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