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
**Problem Statement:**
Solve the following equation for \( x \):
\[ e^{-2x} = \frac{1}{3} \]
**Step-by-Step Solution:**
1. **Take the natural logarithm on both sides of the equation:**
\[ \ln(e^{-2x}) = \ln\left(\frac{1}{3}\right) \]
2. **Simplify the left-hand side using the property of logarithms** \( \ln(e^y) = y \):
\[ -2x = \ln\left(\frac{1}{3}\right) \]
3. **Solve for \( x \)** by dividing both sides by -2:
\[ x = \frac{\ln\left(\frac{1}{3}\right)}{-2} \]
4. **Use the property of logarithms**: \( \ln\left(\frac{1}{a}\right) = -\ln(a) \):
\[ x = \frac{-\ln(3)}{-2} \]
5. **Simplify further to get the final answer:**
\[ x = \frac{\ln(3)}{2} \]
### Conclusion
The value of \( x \) that satisfies the equation \( e^{-2x} = \frac{1}{3} \) is:
\[ x = \frac{\ln(3)}{2} \]
This approach involves using logarithms to linearize the equation and then solving for the variable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F349eaa27-36eb-4309-87e3-9ff3181a3e80%2F43488c2b-ac30-415e-9f04-a01278251777%2F7yr34wc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Exponential Equations
**Problem Statement:**
Solve the following equation for \( x \):
\[ e^{-2x} = \frac{1}{3} \]
**Step-by-Step Solution:**
1. **Take the natural logarithm on both sides of the equation:**
\[ \ln(e^{-2x}) = \ln\left(\frac{1}{3}\right) \]
2. **Simplify the left-hand side using the property of logarithms** \( \ln(e^y) = y \):
\[ -2x = \ln\left(\frac{1}{3}\right) \]
3. **Solve for \( x \)** by dividing both sides by -2:
\[ x = \frac{\ln\left(\frac{1}{3}\right)}{-2} \]
4. **Use the property of logarithms**: \( \ln\left(\frac{1}{a}\right) = -\ln(a) \):
\[ x = \frac{-\ln(3)}{-2} \]
5. **Simplify further to get the final answer:**
\[ x = \frac{\ln(3)}{2} \]
### Conclusion
The value of \( x \) that satisfies the equation \( e^{-2x} = \frac{1}{3} \) is:
\[ x = \frac{\ln(3)}{2} \]
This approach involves using logarithms to linearize the equation and then solving for the variable.
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