To rewrite \(\log(1196)\) using the properties of logarithms, we can first express 1196 in terms of its prime factors. The number 1196 can be factored as follows:
\[ 1196 = 4 \times 299 = 4 \times (300 - 1) \]
However, a more straightforward approach here is to recognize how it can be simplified using the logarithmic properties.
If we take \(1196\) and express it in terms of \(14^2\): \[ 1196 = 14^2 \] We can then rewrite the logarithm: \[ \log(1196) = \log(14^2) \]
Using the Power Rule of Logarithms, which states that \(\log(a^b) = b \cdot \log(a)\), we can express this as: \[ \log(14^2) = 2 \log(14) \]
Thus \( \log(1196) \) can be rewritten as: \[ \log(1196) = 2 \log(14) \]
Now, let's look at the question regarding the option provided.
To find the corresponding answer presented in your response options for this scenario, we need to pay attention to negative coefficients or inverses.
None of the responses are a direct match for \(2 \log(14)\). Instead, we would consider \(2 \log(14)\) as being consistent with the positive nature of the logarithm calculation above.
Therefore, in terms of sign and structured responses, if we were to arrive at similar forms through alternative manipulations, the expression \(−2\log(14)\) presented would NOT correspond correctly.
Yet if you require correlation between negative settings or contextual applications of logarithmic identities through altering base settings or negative placements, it's key to address they derive out of fundamental errors.
Thus, keep note that \(\log(1196)\) is strictly \(\log(14^2)\) or simply \(2 \log(14)\).
Hence the response aligning with your options appears to misformat the entire rewriting structure semantically.
Choose to agree that the accurate interpretation via logarithmic properties yields:
- \( 2 \log(14) \) as a clear relationship re-assertion of logarithm application.