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Let's solve each equation for \( x \).

### 28(a) \( \ln(x^2 - 1) = 3 \)

To solve for \( x \), we will follow these steps:

1. **Isolate the logarithm:**
   \[ \ln(x^2 - 1) = 3 \]

2. **Exponentiate both sides to remove the logarithm:**
   \[ x^2 - 1 = e^3 \]

3. **Solve for \( x \):**
   \[ x^2 = e^3 + 1 \]
   \[ x = \pm \sqrt{e^3 + 1} \]

Thus, the solutions for \( x \) are:
\[ x = \sqrt{e^3 + 1} \quad \text{or} \quad x = -\sqrt{e^3 + 1} \]

### 28(b) \( x^2 - 3e^x + 2 = 0 \)

To solve this equation for \( x \), we will try to find the roots by factoring or using numerical methods.

1. **Rewrite the equation:**
   \[ x^2 - 3e^x + 2 = 0 \]

This equation is not easily factorable, and the term \( e^x \) suggests that it might be transcendental. Therefore, we can try numerical methods or graphical methods to approximate the solutions.

Let's assume we use a numerical method like Newton's method or graphical method to find approximate solutions:

To solve this numerically, we can define a function:
\[ f(x) = x^2 - 3e^x + 2 \]

And find the roots of this function. However, since we do not have computational tools at our disposal here, we can only indicate that numerical methods or a graphing calculator would be required to find approximate solutions.

Would you like more detailed steps or have any questions about solving these equations?

Here are some related questions to expand on this topic:
1. What are some common methods for solving logarithmic equations?
2. How do you solve exponential equations involving \( e \)?
3. What is Newton's method for finding roots of a function?
4. How do you determine the domain and range of a logarithmic function?
5. Can you explain how to graph logarithmic and exponential functions?
6. How do you use a graphing calculator to find the roots of an equation?
7. What are some applications of logarithmic and exponential functions in real life?
8. How do you solve equations that involve both polynomials and exponential terms?

**Tip:** When dealing with equations involving logarithms or exponential functions, it is often useful to use properties of logarithms and exponentials to simplify the equation. For transcendental equations, numerical methods or graphing can be invaluable tools for finding approximate solutions.

Learn how to solve logarithmic and exponential equations step-by-step. Includes detailed methods for isolating logarithms, exponentiating both sides, and using numerical methods for transcendental equations. Suitable for advanced high school students studying algebra and calculus concepts.

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