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To solve the equation:

\[
\ln(2y - 1) = 3
\]

we'll start by isolating \( y \).

### Step 1: Remove the natural logarithm
Rewrite the equation in exponential form to get rid of the \(\ln\) function:
\[
2y - 1 = e^3
\]

### Step 2: Solve for \( y \)
Now, add 1 to both sides:
\[
2y = e^3 + 1
\]

Then, divide both sides by 2:
\[
y = \frac{e^3 + 1}{2}
\]

### Answer
The correct answer is:
\[
y = \frac{e^3 + 1}{2}
\]

Thus, the correct option is:
- \(\displaystyle y = \frac{e^3 + 1}{2}\)

Would you like further details on any step?

Here are 5 related questions to deepen understanding:
1. What is the inverse of a natural logarithm function?
2. How do we transform equations involving logarithms?
3. Why do we use \( e^3 \) instead of \( e \) to the power of other numbers in exponential form?
4. What are the properties of natural logarithms and their applications?
5. How do we simplify expressions like \( \ln(2y - 1) = 3 \)?

**Tip:** When solving logarithmic equations, converting to exponential form can often simplify the solution process.

Step-by-step solution to solve the logarithmic equation ln(2y - 1) = 3. Learn how to isolate y by converting the logarithmic form to an exponential equation. This problem demonstrates key concepts in algebra and logarithmic properties, ideal for high school students in Grades 10-12.

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