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-2x Solve: e²²x = 1/3

-2x Solve: e²²x = 1/3

Algebra and Trigonometry (6th Edition)
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.
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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