Suppose log_k(p) = - 5 and log_k(q) = 3 . 1. The value of log_k(p ^ - 5 * q ^ - 4) is

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.

Learn how to calculate logarithms using properties with a detailed example of finding \(\log_k(p^{-5} \cdot q^{-4})\). This problem involves applying logarithmic properties such as the product rule and exponent rule. Suitable for high school students learning logarithmic functions.

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