To determine which of the given expressions can be rewritten as a single logarithm using the properties of logarithms, let's analyze each option. The properties of logarithms we will use are:
- \( k \cdot \log_b(x) = \log_b(x^k) \) (Power Rule)
- \( \log_b(x) + \log_b(y) = \log_b(x \cdot y) \) (Product Rule)
- \( \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) \) (Quotient Rule)
Analyzing Each Option:
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Option A: \[ \frac{1}{3} \log_2 a - \frac{1}{4} \log_2 b + 2 \log_2 c \]
- Rewrite using the Power Rule: \[ \log_2(a^{1/3}) - \log_2(b^{1/4}) + \log_2(c^2) \]
- Now combine using the Product and Quotient Rules: \[ \log_2\left(\frac{a^{1/3} \cdot c^2}{b^{1/4}}\right) \]
- Thus, this expression can be rewritten as a single logarithm.
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Option B: \[ \frac{1}{3} \log_a \div \frac{1}{4} \log_b \times 2 \log_c \]
- This expression is structured in a way that does not fit directly with logarithmic properties, and it's unclear without additional details how to manipulate it into a single logarithm. It involves division and multiplication of logarithms rather than linear combinations, which do not combine into a single logarithm easily.
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Option C: \[ \frac{1}{3} \log_2 a - \frac{1}{4} \log_3 b + 2 \log_5 c \]
- This choice has logarithms of different bases (2, 3, and 5) which cannot be directly combined into a single logarithm since the base must be the same.
Conclusion:
The expression that can be rewritten as a single logarithm is Option A:
\[ \frac{1}{3} \log_2 a - \frac{1}{4} \log_2 b + 2 \log_2 c. \]