To find the value of $\log_k(p^{-5} \cdot q^{-4})$, we can use the properties of logarithms. The relevant properties are:

1. $\log_k(a \cdot b) = \log_k(a) + \log_k(b)$
2. $\log_k(a^b) = b \cdot \log_k(a)$

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

First, apply the exponent rule to both $p^{-5}$ and $q^{-4}$:

$\log_k(p^{-5}) = -5 \cdot \log_k(p) = -5 \cdot (-5) = 25$

$\log_k(q^{-4}) = -4 \cdot \log_k(q) = -4 \cdot 3 = -12$

Next, use the product rule to combine the two results:

$\log_k(p^{-5} \cdot q^{-4}) = \log_k(p^{-5}) + \log_k(q^{-4}) = 25 + (-12) = 13$

So, the value of $\log_k(p^{-5} \cdot q^{-4})$ is:

$\boxed{13}$

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 $\log_k(p^2)$?
2. How do you express $\log_k(1)$ in terms of $k$?
3. What is $\log_k\left(\frac{1}{p}\right)$?
4. If $k = 10$, what are the values of $\log_{10}(p)$ and $\log_{10}(q)$?
5. What is the inverse function of $\log_k(x)$?
6. How do you simplify $\log_k(pq)$ using $\log_k(p)$ and $\log_k(q)$?
7. How can you express $\log_k(a^m \cdot b^n)$ in terms of $\log_k(a)$ and $\log_k(b)$?
8. If $\log_k(p) = -5$ and $k = 2$, what is the value of $p$?

Tip: When dealing with logarithms, remember that they are the inverse operations of exponentiation. This relationship often helps simplify and solve logarithmic equations.