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### Problem 2: Expanding a logarithmic expression
You are asked to expand the logarithmic expression:

\[
\ln \left( \frac{mn^3}{p} \right)
\]

#### Expansion Process:

1. **Use the logarithmic quotient rule**:
   \[
   \ln \left( \frac{A}{B} \right) = \ln(A) - \ln(B)
   \]
   Applying this to your expression:
   \[
   \ln \left( \frac{mn^3}{p} \right) = \ln(mn^3) - \ln(p)
   \]

2. **Use the logarithmic product rule** for \( \ln(mn^3) \):
   \[
   \ln(AB) = \ln(A) + \ln(B)
   \]
   Therefore:
   \[
   \ln(mn^3) = \ln(m) + \ln(n^3)
   \]

3. **Use the logarithmic power rule** for \( \ln(n^3) \):
   \[
   \ln(n^3) = 3\ln(n)
   \]

   Putting this all together:
   \[
   \ln(mn^3) = \ln(m) + 3\ln(n)
   \]

4. **Combine everything**:
   \[
   \ln \left( \frac{mn^3}{p} \right) = \ln(m) + 3\ln(n) - \ln(p)
   \]

#### Correct Answer:
This matches the third option:
\[
\ln(m) + 3 \ln(n) - \ln(p)
\]

---

### Problem 3: Condensing a logarithmic expression
You are asked to condense the following expression:
\[
3 \ln(x) + 2 \ln(y) + \log_e(z)
\]

#### Condensation Process:

1. **Use the logarithmic power rule** to move the coefficients inside the logs:
   \[
   3 \ln(x) = \ln(x^3), \quad 2 \ln(y) = \ln(y^2)
   \]

2. **Now condense the expression using the logarithmic product rule**:
   \[
   \ln(x^3) + \ln(y^2) + \ln(z) = \ln(x^3 y^2 z)
   \]

#### Correct Answer:
This matches the third option:
\[
\ln(x^3 y^2 z)
\]

---

### Problem 4: Solving a logarithmic equation
You are asked to solve:
\[
\ln(3) + \ln(x) = \ln(24)
\]

#### Solving Process:

1. **Use the logarithmic product rule** on the left-hand side:
   \[
   \ln(3) + \ln(x) = \ln(3x)
   \]

   So, the equation becomes:
   \[
   \ln(3x) = \ln(24)
   \]

2. **Since the logarithms are equal, set the arguments equal to each other**:
   \[
   3x = 24
   \]

3. **Solve for \(x\)**:
   \[
   x = \frac{24}{3} = 8
   \]

#### Correct Answer:
The correct answer is:
\[
x = 8
\]

---

Would you like more details on any of the steps, or further clarification on logarithmic rules?

Here are five additional questions to explore:
1. What are other common logarithmic properties and how do they work?
2. How can logarithms be used to solve exponential equations?
3. What are natural logarithms, and how do they differ from common logarithms?
4. How do you solve logarithmic equations involving multiple terms on each side?
5. What are extraneous solutions in logarithmic equations, and how do you detect them?

**Tip:** When solving logarithmic equations, always check if your solution leads to a negative argument inside the logarithm, as that would indicate an extraneous solution.

Write an equivalent expanded expression: ln((mn^3)/p). Condense the logarithmic expression: 3ln(x) + 2ln(y) + ln(z). Solve ln(3) + ln(x) = ln(24).

Learn how to expand the logarithmic expression ln((mn^3)/p) using logarithmic rules, condense the expression 3ln(x) + 2ln(y) + ln(z), and solve the logarithmic equation ln(3) + ln(x) = ln(24). Detailed step-by-step explanations provided for high school students (Grades 10-12).

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