9) Use the properties of logarithms to write as a single logarithm and make sure all your exponents are positive. 1 4 log(x − 5) + -log(y + 4) − 2log (z + 3)
9) Use the properties of logarithms to write as a single logarithm and make sure all your exponents are positive. 1 4 log(x − 5) + -log(y + 4) − 2log (z + 3)
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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![**Question 9:**
Use the properties of logarithms to write as a single logarithm and make sure all your exponents are positive.
\[ 4 \log(x - 5) + \frac{1}{3} \log(y + 4) - 2 \log(z + 3) \]
**Explanation:**
To combine these logarithms into a single logarithm, apply the properties of logarithms:
1. **Power Rule**: \( a \log(b) = \log(b^a) \)
2. **Product Rule**: \( \log(a) + \log(b) = \log(ab) \)
3. **Quotient Rule**: \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \)
Apply the power rule to each term:
\[ \log((x - 5)^4) + \log((y + 4)^{\frac{1}{3}}) - \log((z + 3)^2) \]
Then, use the product and quotient rules to combine them:
\[ \log\left(\frac{(x - 5)^4 (y + 4)^{\frac{1}{3}}}{(z + 3)^2}\right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F277ae00b-d056-42ed-a729-386f4fca5973%2Fc84ad6c8-fa42-4ed7-9790-a165de420d20%2Fe2gpreb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 9:**
Use the properties of logarithms to write as a single logarithm and make sure all your exponents are positive.
\[ 4 \log(x - 5) + \frac{1}{3} \log(y + 4) - 2 \log(z + 3) \]
**Explanation:**
To combine these logarithms into a single logarithm, apply the properties of logarithms:
1. **Power Rule**: \( a \log(b) = \log(b^a) \)
2. **Product Rule**: \( \log(a) + \log(b) = \log(ab) \)
3. **Quotient Rule**: \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \)
Apply the power rule to each term:
\[ \log((x - 5)^4) + \log((y + 4)^{\frac{1}{3}}) - \log((z + 3)^2) \]
Then, use the product and quotient rules to combine them:
\[ \log\left(\frac{(x - 5)^4 (y + 4)^{\frac{1}{3}}}{(z + 3)^2}\right) \]
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