To solve the equation:

\[
\frac{3}{2} \left( e^{2x} - e^{-2x} \right) = 13 - 3e^{2x}
\]

### Step 1: Multiply through by 2 to eliminate the fraction
Multiply both sides of the equation by 2 to simplify:

\[
3 \left( e^{2x} - e^{-2x} \right) = 2 \left( 13 - 3e^{2x} \right)
\]

This simplifies to:

\[
3 \left( e^{2x} - e^{-2x} \right) = 26 - 6e^{2x}
\]

### Step 2: Expand both sides
Expand the left side:

\[
3e^{2x} - 3e^{-2x} = 26 - 6e^{2x}
\]

### Step 3: Combine like terms
Move all the terms involving \( e^{2x} \) to one side:

\[
3e^{2x} + 6e^{2x} = 26 + 3e^{-2x}
\]

Simplify:

\[
9e^{2x} = 26 + 3e^{-2x}
\]

### Step 4: Multiply by \( e^{2x} \)
Multiply both sides by \( e^{2x} \) to eliminate the \( e^{-2x} \) term:

\[
9e^{4x} = 26e^{2x} + 3
\]

### Step 5: Rearrange into a quadratic form
Rearrange the equation into a quadratic form:

\[
9e^{4x} - 26e^{2x} - 3 = 0
\]

Let \( u = e^{2x} \), so the equation becomes a quadratic in \( u \):

\[
9u^2 - 26u - 3 = 0
\]

### Step 6: Solve the quadratic equation
Solve this quadratic equation using the quadratic formula:

\[
u = \frac{-(-26) \pm \sqrt{(-26)^2 - 4(9)(-3)}}{2(9)}
\]

\[
u = \frac{26 \pm \sqrt{676 + 108}}{18}
\]

\[
u = \frac{26 \pm \sqrt{784}}{18}
\]

\[
u = \frac{26 \pm 28}{18}
\]

Thus, we have two solutions for \( u \):

\[
u = \frac{26 + 28}{18} = \frac{54}{18} = 3
\]

\[
u = \frac{26 - 28}{18} = \frac{-2}{18} = -\frac{1}{9}
\]

### Step 7: Solve for \( x \)
Recall that \( u = e^{2x} \), so:

1. If \( u = 3 \), then \( e^{2x} = 3 \), which gives:

\[
2x = \ln(3)
\]
\[
x = \frac{\ln(3)}{2}
\]

2. If \( u = -\frac{1}{9} \), there is no real solution because \( e^{2x} \) must be positive.

### Final Answer:
The solution is:

\[
x = \frac{\ln(3)}{2}
\]

Would you like any further details or explanations?

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Here are 5 related questions to explore further:
1. How do you solve quadratic equations in different scenarios?
2. What is the significance of logarithms in solving exponential equations?
3. Can you explain the properties of exponential functions?
4. How do we apply the quadratic formula in various contexts?
5. What are common techniques for solving equations with exponential terms?

**Tip**: When solving exponential equations, always check if taking logarithms simplifies the problem!

Learn how to solve the exponential equation 3/2(e^(2x) - e^(-2x)) = 13 - 3e^(2x) step by step. This solution covers key concepts such as quadratic equations, exponential functions, and logarithms. Suitable for students in Grades 10-12.

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