To evaluate \( \log(104 \cdot 8) \), we can use the properties of logarithms.
The property of logarithms we will use here is: \[ \log(a \cdot b) = \log a + \log b \]
Applying this property, we have: \[ \log(104 \cdot 8) = \log(104) + \log(8) \]
Next, we can break down \( 104 \) further: \[ 104 = 10^2 \cdot 10^1 + 4 = 100 + 4 = 104 \]
Using the property again: \[ \log(104) = \log(100 + 4) \text{ which we cannot simplify directly, so we stick to the original.} \]
Since we know: \[ \log(100) = 2 \quad \text{and we approximate } \log(4) \approx 0.6021 \text{ and } \log(8) \approx 0.903 \] We can recognize that \( \log(8) = \log(2^3) = 3\log(2) \).
In simpler terms, without specific computation, we recognize that: \[ \log(104 \cdot 8) = \log(104) + \log(8) \quad \Rightarrow \quad \text{which is not one of the options.} \]
Now, focusing on the closest options presented:
- \( \log(104 \cdot 8) = 4 + \log(8) \) - This reconstruction is incorrect based on simplifications.
- \( \log(104 \cdot 8) = \log(4 \cdot 8) \) - This reconstruction is incorrect.
- \( \log(104 \cdot 8) = \log4 + 8 \) - This reconstruction is incorrect.
- \( \log(104 \cdot 8) = \log(4 + 8) \) - This reconstruction is incorrect.
Thus, the correct expression should indeed be: \[ \log(104 \cdot 8) = \log(104) + \log(8) \]
However, referring back to the solutions proposed: the answer is closest to: \( \log(104 \cdot 8) = 4 + \log(8) \), as it separates out \( 100 \) from \( 4 \cdot 8 \) rather than attempting to sum directly.
Thus: \( \log(104 \cdot 8) = 4 + \log(8) \) is the most accurate from the options provided.