https://file.aiquickdraw.com/mathbot/file/172736544302522hmpxxe.pdf

Let's break down and solve the questions one by one.

### SECTION A: SHORT QUESTIONS
**Question 1.1: Distinguish between the following terms.**
1. **Natural and common logarithms:**
   - **Natural logarithms (ln)** use the base *e* (Euler's number ≈ 2.718). They are written as \( \ln(x) \).
   - **Common logarithms (log)** use the base 10. They are written as \( \log(x) \).

2. **Range and domain:**
   - The **domain** of a function is the set of all possible input values (x-values) for which the function is defined.
   - The **range** is the set of all possible output values (y-values) the function can produce.

---

**Question 1.2: Write \( -4 = \log_{\frac{1}{3}} 81 \) in index form.**

This can be rewritten as:
\[
\frac{1}{3}^{-4} = 81
\]
Thus, the logarithmic form \( \log_{\frac{1}{3}} 81 = -4 \) is correct.

---

**Question 1.3: Write \( n = m^k \) in logarithmic form.**

The logarithmic form of \( n = m^k \) is:
\[
\log_m(n) = k
\]

---

**Question 1.4: Vertical line test for a function (with diagram)**

The vertical line test states that if a vertical line crosses a graph at more than one point, the graph does not represent a function. In a function, each input value (x) must correspond to exactly one output value (y).

---

**Question 1.5: Example of finding the inverse of a function**

To find the inverse of a function, swap the roles of \( x \) and \( y \), and then solve for \( y \). 

For example, for the function \( f(x) = 2x + 3 \):
1. Replace \( f(x) \) with \( y \): \( y = 2x + 3 \)
2. Swap \( x \) and \( y \): \( x = 2y + 3 \)
3. Solve for \( y \): 
   \[
   y = \frac{x - 3}{2}
   \]
   Hence, the inverse function is \( f^{-1}(x) = \frac{x - 3}{2} \).

---

### SECTION B: STRUCTURED QUESTIONS
**Question 1.1: Evaluate \( \log_5 \sqrt{625} \)**

\[
\log_5 \sqrt{625} = \log_5 625^{\frac{1}{2}} = \frac{1}{2} \log_5 625
\]
Since \( 625 = 5^4 \), we have:
\[
\log_5 625 = 4
\]
Thus, 
\[
\log_5 \sqrt{625} = \frac{1}{2} \times 4 = 2
\]

---

**Question 1.2: Expand \( \log_3 \left(\frac{2x^2y^3}{z^4}\right)^2 \)**

\[
\log_3 \left(\frac{2x^2y^3}{z^4}\right)^2 = 2 \log_3 \left(\frac{2x^2y^3}{z^4}\right)
\]
Using logarithm properties:
\[
2 \left[\log_3 2 + \log_3 x^2 + \log_3 y^3 - \log_3 z^4\right]
\]
Simplifying:
\[
2 \left[\log_3 2 + 2 \log_3 x + 3 \log_3 y - 4 \log_3 z\right]
\]
Final answer:
\[
2 \log_3 2 + 4 \log_3 x + 6 \log_3 y - 8 \log_3 z
\]

---

I will continue with the remaining problems next, but let me know if you need further clarifications or if you want to focus on a specific question. Here are some follow-up questions to deepen your understanding:

1. What is the importance of logarithms in mathematical problem-solving?
2. Can you explain the difference between the properties of logarithms and exponents?
3. How do you interpret the inverse of a function geometrically?
4. Why is the vertical line test a reliable method for determining if a graph represents a function?
5. What are common applications of exponential functions in real life?

**Tip:** Always remember to simplify logarithmic expressions step by step using the properties of logarithms.

This page covers the detailed solutions for logarithmic equations, inverse functions, exponential growth calculations, and composition of functions. Concepts include solving logarithmic equations, evaluating inverse functions, and calculating population growth with exponential models. Suitable for advanced high school and college students.

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