Rewrite the log expression as a sum of multiple logs. Further simplify if possible.
$lo{g}_{10}(5\cdot 7)$

• A. ${\log }_{10}5\times {\log }_{10}7$
• B. ${\log }_{10}5+{\log }_{10}7+{\log }_{10}35$
• C. $\(\log\)_{10}5+\(\log\)_{10}7$
• D. $\(\log\)_{10}7-\(\log\)_{10}5$

Answer: (C)

Verified step by step guidance

1. Identify the logarithmic expression given: \(\log_{10}\left(5 \cdot 7\right)\).
2. Recall the logarithm product rule, which states that \(\log_b(xy) = \log_b x + \log_b y\). This allows us to rewrite the log of a product as a sum of logs.
3. Apply the product rule to the expression: \(\log_{10}\left(5 \cdot 7\right) = \log_{10} 5 + \log_{10} 7\).
4. Check if further simplification is possible by evaluating if \(5\) or \(7\) can be broken down into factors with simpler logs, but since both are prime numbers, no further simplification is possible.
5. Conclude that the expression rewritten as a sum of logs is \(\log_{10} 5 + \log_{10} 7\).