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.

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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.

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Question 1.3: Write $n = m^k$ in logarithmic form.

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

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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).

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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 $f^{-1}(x) = \frac{x - 3}{2}$.

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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, $\log_5 \sqrt{625} = \frac{1}{2} \times 4 = 2$

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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: $2 \log_3 2 + 4 \log_3 x + 6 \log_3 y - 8 \log_3 z$

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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.