Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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### 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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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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