e^2x- 6e^x+8=0 Enter your answer as a comma-separated list, and enter none if there are no solutions. 

Let's solve the equation \( e^{2x} - 6e^x + 8 = 0 \).

1. **Substitute** \( u = e^x \), so the equation becomes:
   \[
   u^2 - 6u + 8 = 0
   \]

2. **Solve the quadratic equation**:
   \[
   u^2 - 6u + 8 = 0
   \]
   We can factor this as:
   \[
   (u - 2)(u - 4) = 0
   \]
   So, \( u = 2 \) or \( u = 4 \).

3. **Back-substitute** \( u = e^x \):
   - For \( u = 2 \): \( e^x = 2 \). Taking the natural logarithm of both sides, we get:
     \[
     x = \ln 2
     \]
   - For \( u = 4 \): \( e^x = 4 \). Taking the natural logarithm of both sides, we get:
     \[
     x = \ln 4
     \]

4. **Final solutions**:
   \[
   x = \ln 2, \ln 4
   \]

So, the solution in a comma-separated list is:

\[
\ln 2, \ln 4
\]

Would you like more details or have any questions? Here are some related problems you could consider:

1. Solve \( e^{2x} - 5e^x + 6 = 0 \).
2. Find the solution to \( e^{3x} - 7e^{2x} + 10e^x = 0 \).
3. Solve for \( x \) in the equation \( e^{2x} - 8e^x + 12 = 0 \).
4. Determine the roots of \( e^{x} - 3 = 0 \).
5. What are the solutions to \( e^{2x} - 4e^x = 0 \)?

**Tip:** For equations involving exponentials like \( e^{2x} - 6e^x + 8 = 0 \), substituting \( u = e^x \) simplifies the problem to a quadratic equation, which is easier to solve.

e^2x - 6e^x + 8 = 0. Enter your answer as a comma-separated list, and enter none if there are no solutions.

This guide provides a detailed solution to the exponential equation e^2x - 6e^x + 8 = 0 by using substitution and logarithmic properties. Learn how to factor quadratic equations and apply logarithms for solving exponential problems, suitable for students in Grades 10-12.

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