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Let's analyze the problem.

The expression to evaluate is:

\[
\log(10^4 \cdot 8)
\]

We can simplify this using the properties of logarithms. Specifically, we use the product rule:

\[
\log(a \cdot b) = \log(a) + \log(b)
\]

So, applying this to the expression:

\[
\log(10^4 \cdot 8) = \log(10^4) + \log(8)
\]

Now, we can simplify \(\log(10^4)\) because \(\log(10^4) = 4\log(10)\). Assuming the base of the logarithm is 10, we know \(\log(10) = 1\). Thus:

\[
\log(10^4) = 4
\]

Therefore:

\[
\log(10^4 \cdot 8) = 4 + \log(8)
\]

From the options provided, the correct answer is the first one:

\[
\log(10^4 \cdot 8) = 4 + \log(8)
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How would you evaluate \(\log(10^3 \cdot 5)\)?
2. What is the value of \(\log(100) + \log(50)\)?
3. Can you express \(\log(8)\) in terms of \(\log(2)\)?
4. How can you expand \(\log(200)\) using properties of logarithms?
5. What is \(\log(10^x \cdot y)\) simplified to?

**Tip:** Remember that \(\log(a \cdot b) = \log(a) + \log(b)\) is a key logarithmic property that often simplifies complex expressions!

Learn how to evaluate the logarithmic expression log(10^4 · 8) using the product rule of logarithms. This solution simplifies the expression by applying the key logarithmic property log(a · b) = log(a) + log(b) and calculating log(10^4). Suitable for students in Grades 9-12.

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