https://file.aiquickdraw.com/mathbot/file/1726622156355395nqx2i.jpg

We are given the following information in the image:

\[
\log_7 2 = a \quad \text{and} \quad \log_2 3 = b
\]
We need to find \( \log_6 98 \) in terms of \(a\) and \(b\).

### Solution:

1. **Expressing \( \log_6 98 \) using logarithmic properties:**
   \[
   \log_6 98 = \frac{\log 98}{\log 6}
   \]
2. **Breaking down \( \log 98 \):**
   \[
   98 = 2 \times 49 = 2 \times 7^2
   \]
   Therefore,
   \[
   \log 98 = \log (2 \times 7^2) = \log 2 + 2 \log 7
   \]
3. **Breaking down \( \log 6 \):**
   \[
   6 = 2 \times 3
   \]
   Therefore,
   \[
   \log 6 = \log (2 \times 3) = \log 2 + \log 3
   \]
4. **Substituting known values for \( \log_7 2 = a \) and \( \log_2 3 = b \):**

   From the given information:
   \[
   \log_7 2 = a \implies \log 7 = \frac{\log 2}{a}
   \quad \text{and} \quad
   \log_2 3 = b \implies \log 3 = b \log 2
   \]

   Using these in the equations for \( \log 98 \) and \( \log 6 \):

   \[
   \log 98 = \log 2 + 2 \cdot \frac{\log 2}{a}
   \]
   \[
   \log 6 = \log 2 + b \log 2 = \log 2 (1 + b)
   \]

5. **Final expression for \( \log_6 98 \):**
   \[
   \log_6 98 = \frac{\log 2 + 2 \cdot \frac{\log 2}{a}}{\log 2 (1 + b)}
   \]

   Factor out \( \log 2 \) in both the numerator and the denominator:
   \[
   \log_6 98 = \frac{\log 2 \left( 1 + \frac{2}{a} \right)}{\log 2 (1 + b)} = \frac{1 + \frac{2}{a}}{1 + b}
   \]

Thus, the answer is:
\[
\boxed{ \frac{a + 2}{a(b + 1)} }
\]

So the correct answer is **option C**.

---

Would you like more details or have any further questions? Here are 5 related questions to deepen your understanding:

1. What are the logarithmic properties used to simplify \( \log(2 \times 7^2) \)?
2. How does the change of base formula work for logarithms?
3. How can you derive logarithms in terms of different bases from a given logarithmic relationship?
4. What would \( \log_5 100 \) be if you are given \( \log_5 2 \) and \( \log_5 5 \)?
5. How can logarithms help simplify exponential equations?

**Tip:** Always simplify logarithmic expressions by breaking them down into known components using logarithmic identities.

Jika log base 7 dari 2 sama dengan a dan log base 2 dari 3 sama dengan b, maka log base 6 dari 98 adalah ....

Solve the logarithmic problem involving log base 7 of 2 and log base 2 of 3 to find log base 6 of 98. The solution uses key logarithmic properties and the change of base formula, suitable for Grades 10-12.

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