To solve the equation \( e^{2x} - 1 = 12 \) for \( x \), follow these steps:

### Step 1: Isolate the exponential term
First, add 1 to both sides of the equation to isolate \( e^{2x} \):

\[
e^{2x} = 12 + 1
\]

\[
e^{2x} = 13
\]

### Step 2: Apply the natural logarithm
Take the natural logarithm (ln) of both sides to solve for \( x \):

\[
\ln(e^{2x}) = \ln(13)
\]

Since \( \ln(e^{2x}) = 2x \), the equation simplifies to:

\[
2x = \ln(13)
\]

### Step 3: Solve for \( x \)
Now, divide both sides by 2:

\[
x = \frac{\ln(13)}{2}
\]

Using a calculator to find \( \ln(13) \approx 2.564 \), the solution is:

\[
x \approx \frac{2.564}{2} = 1.282
\]

Thus, \( x \approx 1.282 \).

Would you like further details or clarification?

Here are five related questions for further exploration:

1. How do you solve \( e^x + e^{-x} = 10 \)?
2. What is the derivative of \( f(x) = e^{2x} \)?
3. How do you solve logarithmic equations like \( \ln(x) = 5 \)?
4. Can you solve the equation \( e^{3x} - 2 = 5 \)?
5. How does the natural logarithm function behave as \( x \to \infty \)?

**Tip:** Remember that \( \ln(e^x) = x \), which is a useful property when solving exponential equations.

This guide demonstrates how to solve the exponential equation e^{2x} - 1 = 12 using logarithmic properties. Learn key steps like isolating the exponential term and applying the natural logarithm, suitable for students in Grades 10-12.

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