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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Question
![**Problem 3**
Rewrite \( 3\log_2(a) + 4\log_2(b) - 5\log_2(c) \) as an expression of the form \( \log_2(\text{something}) \).
---
To solve this problem, we will use the properties of logarithms to combine the terms into a single logarithmic expression. Here are the steps involved:
1. **Logarithmic Multiplication Rule**: \( n\log_b(x) = \log_b(x^n) \)
Apply to each term:
\[
3\log_2(a) = \log_2(a^3)
\]
\[
4\log_2(b) = \log_2(b^4)
\]
\[
5\log_2(c) = \log_2(c^5)
\]
2. **Combining Logs**: Using the properties \( \log_b(x) + \log_b(y) = \log_b(xy) \) and \( \log_b(x) - \log_b(y) = \log_b(\frac{x}{y}) \)
Combine the expressions:
\[
\log_2(a^3) + \log_2(b^4) - \log_2(c^5) = \log_2\left(\frac{a^3 \cdot b^4}{c^5}\right)
\]
Thus, the original expression simplifies to:
\[
\log_2\left(\frac{a^3 \cdot b^4}{c^5}\right)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda3861d1-7438-4706-a4d7-876b5456f256%2F9485a129-4cf4-4e46-87e1-29dceab222eb%2Fgu884ol_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3**
Rewrite \( 3\log_2(a) + 4\log_2(b) - 5\log_2(c) \) as an expression of the form \( \log_2(\text{something}) \).
---
To solve this problem, we will use the properties of logarithms to combine the terms into a single logarithmic expression. Here are the steps involved:
1. **Logarithmic Multiplication Rule**: \( n\log_b(x) = \log_b(x^n) \)
Apply to each term:
\[
3\log_2(a) = \log_2(a^3)
\]
\[
4\log_2(b) = \log_2(b^4)
\]
\[
5\log_2(c) = \log_2(c^5)
\]
2. **Combining Logs**: Using the properties \( \log_b(x) + \log_b(y) = \log_b(xy) \) and \( \log_b(x) - \log_b(y) = \log_b(\frac{x}{y}) \)
Combine the expressions:
\[
\log_2(a^3) + \log_2(b^4) - \log_2(c^5) = \log_2\left(\frac{a^3 \cdot b^4}{c^5}\right)
\]
Thus, the original expression simplifies to:
\[
\log_2\left(\frac{a^3 \cdot b^4}{c^5}\right)
\]
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