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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![**Logarithmic Expression Simplification**
**Problem Statement:**
Write the following expression as a logarithm of a single quantity, and simplify when possible:
\[
\frac{3}{5} \log x + \frac{4}{5} \log y
\]
**Detailed Steps to Simplify the Expression:**
1. **Utilize the Power Rule of Logarithms:**
The power rule states that \( a \log b = \log b^a \). We can apply this rule to each term in the expression:
\[
\frac{3}{5} \log x = \log x^{\frac{3}{5}}
\]
\[
\frac{4}{5} \log y = \log y^{\frac{4}{5}}
\]
2. **Combine the Logarithmic Terms Using the Product Rule:**
The product rule of logarithms states that \( \log a + \log b = \log (a \cdot b) \). Apply this rule to combine the terms:
\[
\log x^{\frac{3}{5}} + \log y^{\frac{4}{5}} = \log \left( x^{\frac{3}{5}} \cdot y^{\frac{4}{5}} \right)
\]
**Final Simplified Expression:**
\[
\frac{3}{5} \log x + \frac{4}{5} \log y = \log \left( x^{\frac{3}{5}} \cdot y^{\frac{4}{5}} \right)
\]
By following these steps, the logarithmic expression is simplified into a single quantity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8ec627b-d95d-4a8e-b0e9-8b769944a253%2Fc8d7f695-4271-4cbc-951b-743d2d312b6a%2Fnf8do2a_processed.png&w=3840&q=75)
Transcribed Image Text:**Logarithmic Expression Simplification**
**Problem Statement:**
Write the following expression as a logarithm of a single quantity, and simplify when possible:
\[
\frac{3}{5} \log x + \frac{4}{5} \log y
\]
**Detailed Steps to Simplify the Expression:**
1. **Utilize the Power Rule of Logarithms:**
The power rule states that \( a \log b = \log b^a \). We can apply this rule to each term in the expression:
\[
\frac{3}{5} \log x = \log x^{\frac{3}{5}}
\]
\[
\frac{4}{5} \log y = \log y^{\frac{4}{5}}
\]
2. **Combine the Logarithmic Terms Using the Product Rule:**
The product rule of logarithms states that \( \log a + \log b = \log (a \cdot b) \). Apply this rule to combine the terms:
\[
\log x^{\frac{3}{5}} + \log y^{\frac{4}{5}} = \log \left( x^{\frac{3}{5}} \cdot y^{\frac{4}{5}} \right)
\]
**Final Simplified Expression:**
\[
\frac{3}{5} \log x + \frac{4}{5} \log y = \log \left( x^{\frac{3}{5}} \cdot y^{\frac{4}{5}} \right)
\]
By following these steps, the logarithmic expression is simplified into a single quantity.
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