1. Arun incorrectly writes log, x+log, y -log, z as a single logarithm. log , x+log, y-log, z = log, (x+y-z) Where did Arjun make errors? Explain his errors and the properties oflogarithms that leads to the correct answer. State the correct answer. Answer. (Score for Question 2: of 7 points) 2. Helen and Stephen both simplify the exponential expression Helen: e2-1 ela e Stephen: e n2-1 (2)-1 =e- Un fortunately, they both made an error. Identify where each person made his or her error and explain in words what he or she did wrong. Then, correctly simplify the expression.
1. Arun incorrectly writes log, x+log, y -log, z as a single logarithm. log , x+log, y-log, z = log, (x+y-z) Where did Arjun make errors? Explain his errors and the properties oflogarithms that leads to the correct answer. State the correct answer. Answer. (Score for Question 2: of 7 points) 2. Helen and Stephen both simplify the exponential expression Helen: e2-1 ela e Stephen: e n2-1 (2)-1 =e- Un fortunately, they both made an error. Identify where each person made his or her error and explain in words what he or she did wrong. Then, correctly simplify the expression.
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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![**1.** Arjun incorrectly writes \( \log_7 x + \log_7 y - \log_7 z \) as a single logarithm.
\[
\log_7 x + \log_7 y - \log_7 z = \log_7 (x + y - z)
\]
Where did Arjun make errors? Explain his errors and the properties of logarithms that lead to the correct answer. State the correct answer.
**Answer:**
Arjun made an error by incorrectly applying the properties of logarithms. The correct property for combining logarithms is:
- \(\log_b m + \log_b n = \log_b (mn)\)
- \(\log_b m - \log_b n = \log_b \left(\frac{m}{n}\right)\)
Thus, the correct expression should be:
\[
\log_7 x + \log_7 y - \log_7 z = \log_7 \left(\frac{xy}{z}\right)
\]
**2.** (Score for Question 2: ___ of 7 points)
Helen and Stephen both simplify the exponential expression \( e^{\frac{4}{3}n^2-1} \).
**Helen:**
\[
e^{\frac{4}{3}n^2-1} = \frac{e^{\frac{4}{3}n^2}}{e} = \frac{2^4}{e} = \frac{\sqrt[3]{8}}{e}
\]
**Stephen:**
\[
e^{\frac{4}{3}n^2-1} = e^{3} \cdot e^{-1} = e^3 = e^{\frac{5}{2}} = e^{e\sqrt{2}}
\]
Unfortunately, they both made an error. Identify where each person made his or her error and explain in words what he or she did wrong. Then, correctly simplify the expression.
**Answer:**
- **Helen's Error:** Helen incorrectly interpreted the expression's exponent and simplified it inconsistently with exponential rules. The attempt to express the result as \(\frac{\sqrt[3]{8}}{e}\) implies a misunderstanding of exponent rules.
- **Stephen's Error:** Stephen's work showed the incorrect simplification by treating \(e^{\frac{4}{3}n^2-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ac037ff-948e-417c-8842-ba4baa1f150b%2Fdcf0ad0a-f5ea-4767-a870-da2341fc88e3%2F88754t8_processed.png&w=3840&q=75)
Transcribed Image Text:**1.** Arjun incorrectly writes \( \log_7 x + \log_7 y - \log_7 z \) as a single logarithm.
\[
\log_7 x + \log_7 y - \log_7 z = \log_7 (x + y - z)
\]
Where did Arjun make errors? Explain his errors and the properties of logarithms that lead to the correct answer. State the correct answer.
**Answer:**
Arjun made an error by incorrectly applying the properties of logarithms. The correct property for combining logarithms is:
- \(\log_b m + \log_b n = \log_b (mn)\)
- \(\log_b m - \log_b n = \log_b \left(\frac{m}{n}\right)\)
Thus, the correct expression should be:
\[
\log_7 x + \log_7 y - \log_7 z = \log_7 \left(\frac{xy}{z}\right)
\]
**2.** (Score for Question 2: ___ of 7 points)
Helen and Stephen both simplify the exponential expression \( e^{\frac{4}{3}n^2-1} \).
**Helen:**
\[
e^{\frac{4}{3}n^2-1} = \frac{e^{\frac{4}{3}n^2}}{e} = \frac{2^4}{e} = \frac{\sqrt[3]{8}}{e}
\]
**Stephen:**
\[
e^{\frac{4}{3}n^2-1} = e^{3} \cdot e^{-1} = e^3 = e^{\frac{5}{2}} = e^{e\sqrt{2}}
\]
Unfortunately, they both made an error. Identify where each person made his or her error and explain in words what he or she did wrong. Then, correctly simplify the expression.
**Answer:**
- **Helen's Error:** Helen incorrectly interpreted the expression's exponent and simplified it inconsistently with exponential rules. The attempt to express the result as \(\frac{\sqrt[3]{8}}{e}\) implies a misunderstanding of exponent rules.
- **Stephen's Error:** Stephen's work showed the incorrect simplification by treating \(e^{\frac{4}{3}n^2-
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