Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Solving Equations Using Properties of Exponents and Logarithms
#### Task 1: Solve for \( x \) using the One-to-One Property
**Problem:**
\[ e^{x^2 - 8} = e^{7x} \]
**Implementation:**
Enter your answer in the provided text box and submit.
**Input Field:**
\[ x = \]
\[ \text{Submit Answer} \]
**Explanation:**
By using the One-to-One Property of exponential functions, since \( a^b = a^c \) implies \( b = c \) when \( a > 0 \), solve for \( x \) given the exponential variables equate to each other.
#### Task 2: Solve for \( x \) using the One-to-One Property
**Problem:**
\[ \log_5(x + 1) = \log_5(3) \]
**Implementation:**
Enter your answer in the provided text box as a comma-separated list if there are multiple values.
**Input Field:**
\[ x = \]
**Explanation:**
By using the One-to-One Property of logarithmic functions, since \( \log_b(x) = \log_b(y) \) implies \( x = y \), solve for \( x \).
#### Task 3: Expand the Logarithmic Expression
**Problem:**
\[ \log_6 \left( \frac{x y^2}{z^4} \right) \]
**Implementation:**
Rewrite the logarithmic expression using logarithmic properties and enter the expanded form in the provided text box.
**Input Field:**
Format appropriately to display sums, differences, and/or constant multiples.
**Explanation:**
Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. The properties utilized include:
- \(\log_b (xy) = \log_b (x) + \log_b (y)\)
- \(\log_b \left( \frac{x}{y} \right) = \log_b (x) - \log_b (y)\)
- \(\log_b (x^c) = c \log_b (x)\)
**Note:** Assume all variables are positive to apply these properties correctly.](https://content.bartleby.com/qna-images/question/f2367e66-1ae4-4371-aa7b-db391deab840/b26b8444-4f84-4e19-be4b-47c06edfc599/fn5igw5r_processed.png)
Transcribed Image Text:### Solving Equations Using Properties of Exponents and Logarithms
#### Task 1: Solve for \( x \) using the One-to-One Property
**Problem:**
\[ e^{x^2 - 8} = e^{7x} \]
**Implementation:**
Enter your answer in the provided text box and submit.
**Input Field:**
\[ x = \]
\[ \text{Submit Answer} \]
**Explanation:**
By using the One-to-One Property of exponential functions, since \( a^b = a^c \) implies \( b = c \) when \( a > 0 \), solve for \( x \) given the exponential variables equate to each other.
#### Task 2: Solve for \( x \) using the One-to-One Property
**Problem:**
\[ \log_5(x + 1) = \log_5(3) \]
**Implementation:**
Enter your answer in the provided text box as a comma-separated list if there are multiple values.
**Input Field:**
\[ x = \]
**Explanation:**
By using the One-to-One Property of logarithmic functions, since \( \log_b(x) = \log_b(y) \) implies \( x = y \), solve for \( x \).
#### Task 3: Expand the Logarithmic Expression
**Problem:**
\[ \log_6 \left( \frac{x y^2}{z^4} \right) \]
**Implementation:**
Rewrite the logarithmic expression using logarithmic properties and enter the expanded form in the provided text box.
**Input Field:**
Format appropriately to display sums, differences, and/or constant multiples.
**Explanation:**
Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. The properties utilized include:
- \(\log_b (xy) = \log_b (x) + \log_b (y)\)
- \(\log_b \left( \frac{x}{y} \right) = \log_b (x) - \log_b (y)\)
- \(\log_b (x^c) = c \log_b (x)\)
**Note:** Assume all variables are positive to apply these properties correctly.
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