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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![**Title: Expanding Logarithms Using Log Properties**
---
**Problem Statement:**
Expand the logarithm fully using the properties of logs. Express the final answer in terms of \(\log x\) and \(\log y\).
\[ \log \frac{x}{y^4} \]
**Solution:**
To expand the given logarithm, we will use the following properties of logarithms:
1. **Quotient Rule of Logarithms:**
\[ \log \left(\frac{a}{b}\right) = \log a - \log b \]
2. **Power Rule of Logarithms:**
\[ \log (a^n) = n \log a \]
Applying these rules step-by-step to the given problem:
1. **Apply the Quotient Rule:**
\[ \log \frac{x}{y^4} = \log x - \log y^4 \]
2. **Apply the Power Rule to the second term:**
\[ \log y^4 = 4 \log y \]
Therefore, substituting back:
\[ \log \frac{x}{y^4} = \log x - 4 \log y \]
**Final Expanded Form:**
\[ \log x - 4 \log y \]
This is the expanded form of the given logarithmic expression in terms of \(\log x\) and \(\log y\).
---
This explanation is intended to be clear and comprehensive for students who are learning the properties of logarithms and how to apply them to expand logarithmic expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe45f6d4e-cf07-4f66-9e7e-aaf31f2d53f1%2Ff3ae6992-25ec-4325-9ba5-b19d08141e52%2Fgbosdt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Expanding Logarithms Using Log Properties**
---
**Problem Statement:**
Expand the logarithm fully using the properties of logs. Express the final answer in terms of \(\log x\) and \(\log y\).
\[ \log \frac{x}{y^4} \]
**Solution:**
To expand the given logarithm, we will use the following properties of logarithms:
1. **Quotient Rule of Logarithms:**
\[ \log \left(\frac{a}{b}\right) = \log a - \log b \]
2. **Power Rule of Logarithms:**
\[ \log (a^n) = n \log a \]
Applying these rules step-by-step to the given problem:
1. **Apply the Quotient Rule:**
\[ \log \frac{x}{y^4} = \log x - \log y^4 \]
2. **Apply the Power Rule to the second term:**
\[ \log y^4 = 4 \log y \]
Therefore, substituting back:
\[ \log \frac{x}{y^4} = \log x - 4 \log y \]
**Final Expanded Form:**
\[ \log x - 4 \log y \]
This is the expanded form of the given logarithmic expression in terms of \(\log x\) and \(\log y\).
---
This explanation is intended to be clear and comprehensive for students who are learning the properties of logarithms and how to apply them to expand logarithmic expressions.
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