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!