The question asks us to determine the value of $\log 18$ given that $\log 2 = p$ and $\log 3 = q$.

Solution:

We can use the properties of logarithms and factorize 18: $18 = 2 \cdot 3^2$

Using the logarithmic property: $\log(ab) = \log a + \log b \quad \text{and} \quad \log(a^n) = n \log a,$ we expand $\log 18$: $\log 18 = \log(2 \cdot 3^2) = \log 2 + \log(3^2)$

Simplify further: $\log 18 = \log 2 + 2\log 3$

Substitute $\log 2 = p$ and $\log 3 = q$: $\log 18 = p + 2q$

Answer:

The value of $\log 18$ is $p + 2q$. Hence, the correct option is C.

Do you need more detailed explanations or help with related concepts? Here are some follow-up questions to expand your understanding:

1. What are the key properties of logarithms that simplify calculations like these?
2. How can the logarithmic property $\log(ab) = \log a + \log b$ be applied in other scenarios?
3. What is the significance of base-10 or natural logarithms in real-life applications?
4. Can you rewrite the expression $\log(a^n \cdot b^m)$ in terms of simpler logarithmic expressions?
5. How would you solve $\log 32$ using $\log 2 = p$ without using a calculator?

Tip: Always try to simplify the logarithmic expressions using the properties before substituting values; this often makes solving faster and more organized.