Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Transcription for Educational Website:**
---
**Instruction:**
Use the properties of logarithms to expand the logarithm as much as possible:
\[
4 \log_5 \left( \sqrt[5]{\frac{w^2}{x^4 y^3}} \right)
\]
---
**Step-by-step Expansion:**
1. **Start with the given expression:**
\[
4 \log_5 \left( \sqrt[5]{\frac{w^2}{x^4 y^3}} \right)
\]
2. **Apply the root property of logarithms**:
Convert the fifth root to a power of \(\frac{1}{5}\):
\[
4 \log_5 \left( \left(\frac{w^2}{x^4 y^3}\right)^{\frac{1}{5}} \right)
\]
3. **Use the power rule of logarithms**:
Bring down the exponent \(\frac{1}{5}\):
\[
4 \cdot \frac{1}{5} \log_5 \left(\frac{w^2}{x^4 y^3}\right)
\]
Simplify the constant multiplication:
\[
\frac{4}{5} \log_5 \left(\frac{w^2}{x^4 y^3}\right)
\]
4. **Apply the quotient rule of logarithms**:
Expand using the logarithm of a quotient:
\[
\frac{4}{5} \left(\log_5 (w^2) - \log_5 (x^4 y^3)\right)
\]
5. **Further expand using the product rule of logarithms**:
Separate the logarithm of the product:
\[
\frac{4}{5} \left(\log_5 (w^2) - (\log_5 (x^4) + \log_5 (y^3))\right)
\]
6. **Apply the power rule of logarithms to each term**:
\[
\frac{4}{5} \left(2 \log_5 (w) - (4 \log_5 (x) + 3 \log_5 (y))\right)
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4531c251-9801-4f15-b6f4-822623d05c2f%2F803e4397-1661-4cf6-b13a-9c482d0ffb3d%2Fdbznpjp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
**Instruction:**
Use the properties of logarithms to expand the logarithm as much as possible:
\[
4 \log_5 \left( \sqrt[5]{\frac{w^2}{x^4 y^3}} \right)
\]
---
**Step-by-step Expansion:**
1. **Start with the given expression:**
\[
4 \log_5 \left( \sqrt[5]{\frac{w^2}{x^4 y^3}} \right)
\]
2. **Apply the root property of logarithms**:
Convert the fifth root to a power of \(\frac{1}{5}\):
\[
4 \log_5 \left( \left(\frac{w^2}{x^4 y^3}\right)^{\frac{1}{5}} \right)
\]
3. **Use the power rule of logarithms**:
Bring down the exponent \(\frac{1}{5}\):
\[
4 \cdot \frac{1}{5} \log_5 \left(\frac{w^2}{x^4 y^3}\right)
\]
Simplify the constant multiplication:
\[
\frac{4}{5} \log_5 \left(\frac{w^2}{x^4 y^3}\right)
\]
4. **Apply the quotient rule of logarithms**:
Expand using the logarithm of a quotient:
\[
\frac{4}{5} \left(\log_5 (w^2) - \log_5 (x^4 y^3)\right)
\]
5. **Further expand using the product rule of logarithms**:
Separate the logarithm of the product:
\[
\frac{4}{5} \left(\log_5 (w^2) - (\log_5 (x^4) + \log_5 (y^3))\right)
\]
6. **Apply the power rule of logarithms to each term**:
\[
\frac{4}{5} \left(2 \log_5 (w) - (4 \log_5 (x) + 3 \log_5 (y))\right)
\
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