Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Solving Exponential Equations Using Logarithms
**Problem:**
\[ \text{Solve by using logarithms:} \]
\[ 2^{x+3} = 5^{2-x} \]
**Solution:**
To solve this equation, we need to use logarithms to isolate the variable \( x \). Here are the steps:
1. **Take the natural logarithm (ln) of both sides:**
\[ \ln(2^{x+3}) = \ln(5^{2-x}) \]
2. **Apply the power rule of logarithms** \( \ln(a^b) = b \cdot \ln(a) \):
\[ (x+3) \cdot \ln(2) = (2-x) \cdot \ln(5) \]
3. **Distribute the logarithms:**
\[ x \ln(2) + 3 \ln(2) = 2 \ln(5) - x \ln(5) \]
4. **Collect the terms involving \( x \) on one side:**
\[ x \ln(2) + x \ln(5) = 2 \ln(5) - 3 \ln(2) \]
5. **Factor out \( x \) from the left-hand side:**
\[ x (\ln(2) + \ln(5)) = 2 \ln(5) - 3 \ln(2) \]
6. **Solve for \( x \):**
\[ x = \frac{2 \ln(5) - 3 \ln(2)}{\ln(2) + \ln(5)} \]
This equation now gives us the value of \( x \) in terms of natural logarithms.
**Explanation of Concepts:**
- **Natural Logarithm (ln):** The natural logarithm is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental constant approximately equal to 2.718281828459.
- **Power Rule of Logarithms:** This rule allows us to move the exponent in front of the logarithm, simplifying the equation.
By solving step-by-step using logarithmic properties, we've isolated \( x \) in the equation \( 2^{x+3} = 5^{2-x} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2c5c293-7ff8-490b-b548-13e2f43ca7a8%2Fce736ff0-e66b-4727-97b2-87e74a9c308e%2F0rm9a68.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Exponential Equations Using Logarithms
**Problem:**
\[ \text{Solve by using logarithms:} \]
\[ 2^{x+3} = 5^{2-x} \]
**Solution:**
To solve this equation, we need to use logarithms to isolate the variable \( x \). Here are the steps:
1. **Take the natural logarithm (ln) of both sides:**
\[ \ln(2^{x+3}) = \ln(5^{2-x}) \]
2. **Apply the power rule of logarithms** \( \ln(a^b) = b \cdot \ln(a) \):
\[ (x+3) \cdot \ln(2) = (2-x) \cdot \ln(5) \]
3. **Distribute the logarithms:**
\[ x \ln(2) + 3 \ln(2) = 2 \ln(5) - x \ln(5) \]
4. **Collect the terms involving \( x \) on one side:**
\[ x \ln(2) + x \ln(5) = 2 \ln(5) - 3 \ln(2) \]
5. **Factor out \( x \) from the left-hand side:**
\[ x (\ln(2) + \ln(5)) = 2 \ln(5) - 3 \ln(2) \]
6. **Solve for \( x \):**
\[ x = \frac{2 \ln(5) - 3 \ln(2)}{\ln(2) + \ln(5)} \]
This equation now gives us the value of \( x \) in terms of natural logarithms.
**Explanation of Concepts:**
- **Natural Logarithm (ln):** The natural logarithm is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental constant approximately equal to 2.718281828459.
- **Power Rule of Logarithms:** This rule allows us to move the exponent in front of the logarithm, simplifying the equation.
By solving step-by-step using logarithmic properties, we've isolated \( x \) in the equation \( 2^{x+3} = 5^{2-x} \).
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