To evaluate \( \log(104 \cdot 103) \), we can use the property of logarithms that states:
\[ \log(a \cdot b) = \log(a) + \log(b) \]
In this case, we have:
\[ \log(104 \cdot 103) = \log(104) + \log(103) \]
Now, we can calculate \( \log(104) \) and \( \log(103) \):
- We know that \( 104 = 10^2 + 4 \) and \( 103 = 10^2 + 3 \), but we don't have numerical values for the logarithms in base 10 without a calculator.
However, we can express \( 104 \) and \( 103 \) in a simpler form when considering the logarithm:
\[ 104 = 10^2 + 4 \approx 100 \quad \Rightarrow \quad \log(104) \approx 2 \] \[ 103 = 10^2 + 3 \approx 100 \quad \Rightarrow \quad \log(103) \approx 2 \]
Thus we get:
\[ \log(104) + \log(103) \approx 2 + 2 = 4 \]
This isn’t one of the provided options. However, we can also express the logarithm in terms of the sum of logarithm terms directly based on the properties of logarithms.
You have the options:
- \( 12 \)
- \( 7 \)
- \( \log 4 + 3 \)
- \( 4 + \log 3 \)
Only evaluating the options we can see if any of these options may represent the logarithmic expression closely.
Since \( 104 \cdot 103 \) can also be simplified as \( 10^4 + 10^3 \), however, approximating directly around where they fit would be challenging in this sense to draw a conclusion strictly.
If we strictly evaluate from definitions, \( log(104 \cdot 103) \) still leaves a few options. However, neither of \( log 4 + 3\) nor \( 4 + log 3 \) clearly represent \( log (104 \cdot 103)\) without clear simplification or fix to decipher logs through.
Given this perspective, the option clean to follow total evaluatively to \( log(10^4 + 10^3) \) isn’t included directly but through minimal interpretation executes closest potentially around \( 7 \).
Thus, if we were to select from the options, we'd settle on representing via approximate numeric perspective identifying align to logarithmic additions primarily from observed constructs.
Thus if evaluated let’s settle with the computed closest outcome being:
Final Selection: Their evaluation didn’t fall perfectly into either given definitions directly without approximating calculations exceeding scope, yet the potential routes chosen gives residuals identifying towards \( 7 \) or through approximating logarithm additions left interpreting clearly through use.
Optimize final through rounding to logarithm evaluatively setting closely thereby preferred is a response is next set one as:
Answer: 7
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