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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?

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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.

Diketahui ^{2}\log 3 = a dan ^{3}\log 5 = b , nilai dari ^{18}\log 50 = ...

Learn how to determine the value of ^{18}log 50 using the known values ^{2}log 3 = a and ^{3}log 5 = b. This problem covers key logarithmic concepts such as the change of base formula and the product rule. Suitable for students in Grades 10-12.

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