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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![**Consolidating Logarithmic Expressions**
In this section, we will explore how to write the expression \(2\ln(8) + 5\ln(z)\) as a single logarithm.
---
**Question:**
*Write the following expression \(2\ln(8) + 5\ln(z)\) as a single logarithm.*
**Options:**
- **(A)** \(\ln \left(\frac{64}{z^5}\right)\)
- **(B)** \(\ln(8z^5)\)
- **(C)** \(\frac{2}{5}\ln \left(8z\right)\)
- **(D)** \(\ln(64z^5)\)
---
To consolidate the logarithmic expression \(2\ln(8) + 5\ln(z)\) into a single logarithm, we can use the properties of logarithms:
1. **Power Rule:** \(a \ln(x) = \ln(x^a)\)
2. **Product Rule:** \(\ln(a) + \ln(b) = \ln(ab)\)
Applying the power rule:
\[ 2\ln(8) = \ln(8^2) = \ln(64) \]
\[ 5\ln(z) = \ln(z^5) \]
Next, applying the product rule to combine \(\ln(64)\) and \(\ln(z^5)\):
\[ \ln(64) + \ln(z^5) = \ln(64z^5) \]
Thus, the correct single logarithm expression is:
\[ \ln(64z^5) \]
So, the correct answer from the given options is:
- **Option (D):** \(\ln(64z^5)\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85b5253c-003b-4a21-966f-94782a914797%2F6033e9c9-fd67-4d6b-aff2-2efe0fce12e3%2Fn0jcra_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consolidating Logarithmic Expressions**
In this section, we will explore how to write the expression \(2\ln(8) + 5\ln(z)\) as a single logarithm.
---
**Question:**
*Write the following expression \(2\ln(8) + 5\ln(z)\) as a single logarithm.*
**Options:**
- **(A)** \(\ln \left(\frac{64}{z^5}\right)\)
- **(B)** \(\ln(8z^5)\)
- **(C)** \(\frac{2}{5}\ln \left(8z\right)\)
- **(D)** \(\ln(64z^5)\)
---
To consolidate the logarithmic expression \(2\ln(8) + 5\ln(z)\) into a single logarithm, we can use the properties of logarithms:
1. **Power Rule:** \(a \ln(x) = \ln(x^a)\)
2. **Product Rule:** \(\ln(a) + \ln(b) = \ln(ab)\)
Applying the power rule:
\[ 2\ln(8) = \ln(8^2) = \ln(64) \]
\[ 5\ln(z) = \ln(z^5) \]
Next, applying the product rule to combine \(\ln(64)\) and \(\ln(z^5)\):
\[ \ln(64) + \ln(z^5) = \ln(64z^5) \]
Thus, the correct single logarithm expression is:
\[ \ln(64z^5) \]
So, the correct answer from the given options is:
- **Option (D):** \(\ln(64z^5)\)
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