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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![**Task: Simplify to a Single Logarithm**
Given Expression:
\[ 3\ln 2 - \frac{1}{2}\ln(x-1) + 2\ln(3x+1) + \frac{2}{3}\ln(x^2+9) - 2\ln 3 - \ln x \]
**Steps:**
1. **Use Logarithm Power Rule:**
\[
a\ln(b) = \ln(b^a)
\]
Apply transformations:
- \( \ln(2^3) = \ln(8) \)
- \( \ln((x-1)^{1/2}) = \ln(\sqrt{x-1}) \)
- \( \ln((3x+1)^2) \)
- \( \ln((x^2+9)^{2/3}) = \ln(\sqrt[3]{(x^2+9)^2}) \)
- \( \ln(3^2) = \ln(9) \)
2. **Combine the Logarithms:**
Apply the logarithm sum and difference rules:
\[
\ln(a) + \ln(b) = \ln(ab)
\]
\[
\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)
\]
Formulate into a single logarithmic expression:
\[
\ln\left(\frac{8(3x+1)^2\sqrt[3]{(x^2+9)^2}}{\sqrt{x-1} \cdot 9x}\right)
\]
**Final Expression:**
The entire expression simplifies to a single logarithm:
\[
\ln\left(\frac{8(3x+1)^2\sqrt[3]{(x^2+9)^2}}{\sqrt{x-1} \cdot 9x}\right)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F153e7a04-5e97-418e-a56b-c262c9ac626d%2Fed0aa892-bde0-45d6-8918-bab96c6521e1%2Fmkvcx1q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task: Simplify to a Single Logarithm**
Given Expression:
\[ 3\ln 2 - \frac{1}{2}\ln(x-1) + 2\ln(3x+1) + \frac{2}{3}\ln(x^2+9) - 2\ln 3 - \ln x \]
**Steps:**
1. **Use Logarithm Power Rule:**
\[
a\ln(b) = \ln(b^a)
\]
Apply transformations:
- \( \ln(2^3) = \ln(8) \)
- \( \ln((x-1)^{1/2}) = \ln(\sqrt{x-1}) \)
- \( \ln((3x+1)^2) \)
- \( \ln((x^2+9)^{2/3}) = \ln(\sqrt[3]{(x^2+9)^2}) \)
- \( \ln(3^2) = \ln(9) \)
2. **Combine the Logarithms:**
Apply the logarithm sum and difference rules:
\[
\ln(a) + \ln(b) = \ln(ab)
\]
\[
\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)
\]
Formulate into a single logarithmic expression:
\[
\ln\left(\frac{8(3x+1)^2\sqrt[3]{(x^2+9)^2}}{\sqrt{x-1} \cdot 9x}\right)
\]
**Final Expression:**
The entire expression simplifies to a single logarithm:
\[
\ln\left(\frac{8(3x+1)^2\sqrt[3]{(x^2+9)^2}}{\sqrt{x-1} \cdot 9x}\right)
\]
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