To solve the equation \(3e^{2x - 1} = 15\), we will first isolate the exponential term and then apply the natural logarithm to solve for \(x\).

### Step-by-Step Solution

1. **Isolate the exponential term:**

   \[
   e^{2x - 1} = \frac{15}{3} = 5
   \]

2. **Apply the natural logarithm (ln) to both sides:**

   \[
   \ln\left(e^{2x - 1}\right) = \ln(5)
   \]

   Since \(\ln(e^y) = y\), this simplifies to:

   \[
   2x - 1 = \ln(5)
   \]

3. **Solve for \(x\):**

   \[
   2x = \ln(5) + 1
   \]

   \[
   x = \frac{\ln(5) + 1}{2}
   \]

4. **Calculate the numerical value (optional):**

   Using \(\ln(5) \approx 1.6094\):

   \[
   x \approx \frac{1.6094 + 1}{2} = \frac{2.6094}{2} \approx 1.3047
   \]

### Final Answer

\[
x = \frac{\ln(5) + 1}{2} \approx 1.3047
\]

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How do we solve exponential equations with different bases?
2. What is the derivative of the function \(e^{2x - 1}\)?
3. How can we solve logarithmic equations?
4. What are the properties of logarithms used in solving equations?
5. How do we solve exponential equations involving multiple exponential terms?

**Tip:** When solving exponential equations, isolating the exponential expression and using logarithms are key steps.

Learn how to solve the equation 3e^(2x - 1) = 15 with detailed steps and calculations. Understand exponential function properties and the use of natural logarithms in solving equations. Suitable for advanced high school students and anyone interested in advanced algebraic techniques.

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