https://file.aiquickdraw.com/mathbot/file/17254675438007de6jxvu.jpg

The problem in the image is:

\[
\log \left(10^a \times 100\right)
\]

To solve this, we'll use the logarithmic properties:

1. **Product Property**: \(\log (xy) = \log x + \log y\)
2. **Power Rule**: \(\log (x^y) = y \log x\)
3. **Logarithm of a Power of 10**: \(\log(10^n) = n\)

### Step-by-Step Solution:

1. **Expand using the product property**:
   \[
   \log \left(10^a \times 100\right) = \log(10^a) + \log(100)
   \]

2. **Apply the power rule** to the first term:
   \[
   \log(10^a) = a \cdot \log(10)
   \]
   Since \(\log(10) = 1\), this simplifies to:
   \[
   \log(10^a) = a
   \]

3. **Simplify the second term**:
   \[
   \log(100) = \log(10^2) = 2 \cdot \log(10) = 2
   \]

4. **Combine the results**:
   \[
   \log \left(10^a \times 100\right) = a + 2
   \]

### Final Answer:
\[
\log \left(10^a \times 100\right) = a + 2
\]

Would you like more details or have any questions?

---

Here are five questions related to this problem:
1. What would be the result if \(a = 3\) in the expression \(\log \left(10^a \times 100\right)\)?
2. How does the logarithm of a product differ from the logarithm of a quotient?
3. Can you express \( \log(1000) \) in terms of \(a\) if \(1000 = 10^a\)?
4. What happens to the expression if \(a = -2\)?
5. How can the logarithm properties simplify solving exponential equations?

**Tip:** Remember that \(\log_b(10^x) = x\) when the base \(b\) of the logarithm is 10.

This problem involves solving the logarithmic expression log(10^a × 100) using key logarithmic properties such as the product property and power rule. Suitable for students in Grades 9-12.

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