The problem provided is:

$^{2}\log 3 = a \text{ and } ^{3}\log 5 = b \text{, determine the value of } ^{18}\log 50.$

Let's break this down:

Step 1: Express $^{18}\log 50$

We know that:

$50 = 2 \times 5^2$

Using the logarithmic property:

$^{18}\log 50 = ^{18}\log (2 \times 5^2)$

This can be split into two terms using the product rule for logarithms:

$^{18}\log 50 = ^{18}\log 2 + 2 \cdot ^{18}\log 5$

Step 2: Change of base formula

The change of base formula is:

$^{m}\log n = \frac{\log n}{\log m}$

For $^{18}\log 2$:

$^{18}\log 2 = \frac{\log 2}{\log 18}$

For $^{18}\log 5$:

$^{18}\log 5 = \frac{\log 5}{\log 18}$

Step 3: Substituting Known Values

From the question, we know:

• $^{2}\log 3 = a$, which means $\frac{\log 3}{\log 2} = a$.
• $^{3}\log 5 = b$, which means $\frac{\log 5}{\log 3} = b$.

These relationships can be used to compute $^{18}\log 50$, but we'd need to know more about $^{18}\log 3$ and $^{18}\log 5$. Would you like to explore this further?

Here are five follow-up questions to consider:

1. What logarithmic property can help simplify $^{m}\log (a \times b)$?
2. How do you apply the change of base formula to other logarithms?
3. Can we solve $^{2}\log 3 = a$ to find a more precise value for $\log 3$?
4. What additional information would help solve this problem more efficiently?
5. How does the change of base formula impact different logarithmic values?

Tip: When working with logarithms, remember that properties like the product rule, quotient rule, and change of base formula are your best tools for simplification.