The Product Rule of Logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. In mathematical terms, this is expressed as:
\[ \log_b(M \cdot N) = \log_b(M) + \log_b(N) \]
To apply this to the expression \( \log_{24}(M) \), we need to express \( 24 \) as a product of two numbers:
\[ 24 = 3 \cdot 8 \]
Using the Product Rule, we can rewrite \( \log_{24}(24) \) as:
\[ \log_{24}(24) = \log_{24}(3 \cdot 8) = \log_{24}(3) + \log_{24}(8) \]
None of the provided options directly correspond to the Product Rule applied to \( \log_{24}(3 \cdot 8) \). However, if the intention was to find \( \log_{24}(3) + \log_{24}(8) \) without specifying that the logarithm is of 24, this matches with the format of the option:
- \( \log 3 + \log 8 \)
So the correct response that indicates an equivalent expression using the Product Rule is:
log3 + log8 (the first response).
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