Part 1 : Suppose log_k(p) = - 5 and log_k(q) = 3 . 1. The value of log_k(p ^ - 5 * q ^ - 4) is 13 2. The value of log_k(root(p ^ 2 * q ^ 5, 2)) is 3. One way to express k ^ - 9 in terms of p and is in terms of k ^ - 9 = 4. If log_k(r) = 16 , one way to express r in terms of p and q is

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.

Explore how to solve logarithmic equations involving exponents step-by-step. This includes calculating complex logarithmic expressions and expressing variables in terms of logarithmic values. Suitable for high school students learning advanced algebra concepts.

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