Solve The Equation:${ E^{-x} = 1 - E^{-x} }$

Introduction

In mathematics, equations involving exponential functions can be challenging to solve. The equation $e^{-x} = 1 - e^{-x}$ is a classic example of such an equation. In this article, we will explore the steps to solve this equation and provide a clear understanding of the solution process.

Understanding the Equation

The given equation is $e^{-x} = 1 - e^{-x}$. This equation involves an exponential function with a negative exponent, which can be rewritten as $e^{-x} = e^{-x} - 1$. To simplify the equation, we can add $e^{-x}$ to both sides, resulting in $2e^{-x} = 1$.

Solving for $x$

To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can start by dividing both sides of the equation by $2$, resulting in $e^{-x} = \frac{1}{2}$. Next, we can take the natural logarithm (ln) of both sides, which gives us $-x = \ln\left(\frac{1}{2}\right)$.

Simplifying the Equation

To simplify the equation further, we can multiply both sides by $-1$, resulting in $x = -\ln\left(\frac{1}{2}\right)$. This equation can be rewritten as $x = \ln\left(\frac{1}{2}\right)^{-1}$, using the property of logarithms that states $\ln(a^{-1}) = -\ln(a)$.

Evaluating the Solution

To evaluate the solution, we can use a calculator to find the value of $x$. Plugging in the value of $\ln\left(\frac{1}{2}\right)$, we get $x \approx 0.693147$. This means that the solution to the equation $e^{-x} = 1 - e^{-x}$ is $x \approx 0.693147$.

Conclusion

In this article, we have solved the equation $e^{-x} = 1 - e^{-x}$ using algebraic manipulations and logarithmic properties. The solution to the equation is $x = \ln\left(\frac{1}{2}\right)^{-1}$, which can be evaluated to find the approximate value of $x$. This equation is a classic example of an exponential equation, and the solution process provides a clear understanding of how to approach such equations.

Additional Examples

Here are a few additional examples of exponential equations that can be solved using similar techniques:

• $e^{2x} = 4e^{x}$
• $e^{-3x} = \frac{1}{8}$
• $e^{x} = 2e^{-x}$

These examples can be solved using the same steps as the original equation, and the solutions can be evaluated to find the approximate values of $x$.

Tips and Tricks

Here are a few tips and tricks for solving exponential equations:

• Use algebraic manipulations: Exponential equations can often be simplified using algebraic manipulations, such as adding or subtracting terms.
• Use logarithmic properties: Logarithmic properties, such as the property that $\ln(a^{-1}) = -\ln(a)$, can be used to simplify exponential equations.
• Use a calculator: A calculator can be used to evaluate the solution to an exponential equation.

By following these tips and tricks, you can solve exponential equations with ease and confidence.

Final Thoughts

Solving exponential equations can be a challenging task, but with the right techniques and tools, it can be done. In this article, we have solved the equation $e^{-x} = 1 - e^{-x}$ using algebraic manipulations and logarithmic properties. The solution to the equation is $x = \ln\left(\frac{1}{2}\right)^{-1}$, which can be evaluated to find the approximate value of $x$. We hope that this article has provided a clear understanding of how to approach exponential equations and has given you the confidence to tackle more challenging problems.

Introduction

In our previous article, we solved the equation $e^{-x} = 1 - e^{-x}$ using algebraic manipulations and logarithmic properties. In this article, we will answer some frequently asked questions (FAQs) related to the solution of this equation.

Q&A

Q: What is the main concept behind solving the equation $e^{-x} = 1 - e^{-x}$?

A: The main concept behind solving the equation $e^{-x} = 1 - e^{-x}$ is to use algebraic manipulations and logarithmic properties to isolate the variable $x$ on one side of the equation.

Q: How do I simplify the equation $e^{-x} = 1 - e^{-x}$?

A: To simplify the equation $e^{-x} = 1 - e^{-x}$, you can add $e^{-x}$ to both sides, resulting in $2e^{-x} = 1$. Then, you can divide both sides by $2$, resulting in $e^{-x} = \frac{1}{2}$.

Q: What is the next step after simplifying the equation $e^{-x} = \frac{1}{2}$?

A: After simplifying the equation $e^{-x} = \frac{1}{2}$, you can take the natural logarithm (ln) of both sides, resulting in $-x = \ln\left(\frac{1}{2}\right)$. Then, you can multiply both sides by $-1$, resulting in $x = -\ln\left(\frac{1}{2}\right)$.

Q: How do I evaluate the solution $x = -\ln\left(\frac{1}{2}\right)$?

A: To evaluate the solution $x = -\ln\left(\frac{1}{2}\right)$, you can use a calculator to find the value of $\ln\left(\frac{1}{2}\right)$. Then, you can multiply the result by $-1$ to find the value of $x$.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not using algebraic manipulations to simplify the equation
• Not using logarithmic properties to isolate the variable $x$
• Not evaluating the solution correctly using a calculator

Q: Can you provide some additional examples of exponential equations that can be solved using similar techniques?

A: Yes, here are a few additional examples of exponential equations that can be solved using similar techniques:

• $e^{2x} = 4e^{x}$
• $e^{-3x} = \frac{1}{8}$
• $e^{x} = 2e^{-x}$

These examples can be solved using the same steps as the original equation, and the solutions can be evaluated to find the approximate values of $x$.

Q: What are some tips and tricks for solving exponential equations?

A: Some tips and tricks for solving exponential equations include:

• Using algebraic manipulations to simplify the equation
• Using logarithmic properties to isolate the variable $x$
• Using a calculator to evaluate the solution

By following these tips and tricks, you can solve exponential equations with ease and confidence.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to the solution of the equation $e^{-x} = 1 - e^{-x}$. We hope that this article has provided a clear understanding of how to approach exponential equations and has given you the confidence to tackle more challenging problems.

Final Thoughts

Solving exponential equations can be a challenging task, but with the right techniques and tools, it can be done. In this article, we have provided a clear understanding of how to approach exponential equations and have given you the confidence to tackle more challenging problems. We hope that this article has been helpful and informative, and we look forward to hearing from you in the future.