Let's solve each part step-by-step.

Part 1

1. Given: $\log_k(p) = -5 \quad \text{and} \quad \log_k(q) = 3$ Find: $\log_k(p^{-5} \cdot q^{-4}) = 13$

   This has already been calculated as 13.

Part 2

2. Given: $\log_k(p) = -5 \quad \text{and} \quad \log_k(q) = 3$ Find: $\log_k\left(\sqrt{p^2 \cdot q^5}\right)$

   First, we simplify the expression inside the logarithm: $\sqrt{p^2 \cdot q^5} = (p^2 \cdot q^5)^{1/2}$

   Apply the power rule: $\log_k\left((p^2 \cdot q^5)^{1/2}\right) = \frac{1}{2} \log_k(p^2 \cdot q^5)$

   Use the product rule: $\log_k(p^2 \cdot q^5) = \log_k(p^2) + \log_k(q^5)$

   Apply the power rule to each term: $\log_k(p^2) = 2 \log_k(p) = 2 \cdot (-5) = -10$ $\log_k(q^5) = 5 \log_k(q) = 5 \cdot 3 = 15$

   Combine the terms: $\log_k(p^2 \cdot q^5) = -10 + 15 = 5$

   Now apply the $\frac{1}{2}$ factor: $\frac{1}{2} \log_k(p^2 \cdot q^5) = \frac{1}{2} \cdot 5 = 2.5$

   So the value is: $\log_k\left(\sqrt{p^2 \cdot q^5}\right) = 2.5$

Part 3

3. Expressing $k^{-9}$ in terms of $p$ and $q$:

   We know that: $\log_k(p) = -5 \quad \text{and} \quad \log_k(q) = 3$

   First, let's express $k^{-9}$ in terms of $k$: $k^{-9} = \frac{1}{k^9}$

   Using the definitions of logarithms, express $p$ and $q$ in terms of $k$: $p = k^{-5} \quad \text{and} \quad q = k^3$

   Let's find an expression for $k^{-9}$: $k^{-9} = (k^{-5})^{9/5} = p^{9/5}$ $k^{-9} = (k^{3})^{-3} = q^{-3}$

   So, we have: $k^{-9} = p^{9/5} \quad \text{or} \quad k^{-9} = q^{-3}$

Part 4

4. Given: $\log_k(r) = 16$ Express $r$ in terms of $p$ and $q$:

   We know: $\log_k(p) = -5 \quad \text{and} \quad \log_k(q) = 3$

   From the logarithm definitions: $p = k^{-5} \quad \text{and} \quad q = k^3$

   We need to find $r$ such that: $\log_k(r) = 16 \implies r = k^{16}$

   Let's express $k^{16}$ in terms of $p$ and $q$: $k^{16} = (k^{-5})^{-16/5} = p^{-16/5}$ $k^{16} = (k^3)^{16/3} = q^{16/3}$

   So, we have: $r = p^{-16/5} \quad \text{or} \quad r = q^{16/3}$

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:

1. What is the value of $\log_k(p^3 \cdot q^2)$?
2. How do you simplify $\log_k(\frac{p}{q})$ using $\log_k(p)$ and $\log_k(q)$?
3. What is the inverse function of $\log_k(x)$?
4. How can you express $k^7$ in terms of $p$ and $q$?
5. If $\log_k(s) = 2$, how can $s$ be expressed in terms of $p$ and $q$?
6. What is $\log_k(p^4)$?
7. How do you simplify $\log_k(p^3 \cdot q^{-2})$?
8. If $\log_k(t) = -1$, how can $t$ be expressed in terms of $p$ and $q$?

Tip: Remember that logarithms and exponents are inverse operations. This relationship is key to transforming expressions and solving logarithmic equations.