We are given the equation:

\[
e^x = 2e^{-x}
\]

To solve this, let's first eliminate the negative exponent by multiplying both sides of the equation by \( e^x \). This will simplify the equation and help us isolate \( x \).

### Step 1: Multiply both sides by \( e^x \)
\[
e^x \cdot e^x = 2e^{-x} \cdot e^x
\]
This simplifies to:
\[
e^{2x} = 2
\]

### Step 2: Take the natural logarithm of both sides
Now, take the natural logarithm (ln) of both sides to solve for \( x \):
\[
\ln(e^{2x}) = \ln(2)
\]
Using the property of logarithms \( \ln(e^a) = a \), this simplifies to:
\[
2x = \ln(2)
\]

### Step 3: Solve for \( x \)
Now, divide both sides by 2 to isolate \( x \):
\[
x = \frac{\ln(2)}{2}
\]

Thus, the solution is:
\[
x = \frac{\ln(2)}{2}
\]

Would you like more details or have any questions about the steps? 

Here are some related questions to explore further:
1. What are some other methods to solve exponential equations?
2. How does the natural logarithm simplify the process in this case?
3. What is the numerical value of \( x \)?
4. How can we use this solution in real-world applications?
5. Can we solve equations involving other exponential functions, such as \( 3e^x = 5e^{-x} \)?

**Tip:** When solving exponential equations, multiplying both sides by the same exponential expression can often simplify terms and help you isolate the variable efficiently.

Learn how to solve the exponential equation e^x = 2e^-x step-by-step. This solution covers key algebraic concepts, including exponential equations and logarithmic properties. Suitable for students in Grades 9-11.

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