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.