To determine which expressions can be rewritten as a single logarithm, we can use the properties of logarithms:
- Product Rule: \(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\)
- Quotient Rule: \(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\)
- Power Rule: \(k \cdot \log_b(m) = \log_b(m^k)\)
Now let's analyze each of the provided expressions:
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Expression: \(13\log_2(a) - 14\log_2(b) + 2\log_2(c)\)
- This can be rewritten using the properties of logarithms:
- First, apply the Power Rule:
- \(13\log_2(a) = \log_2(a^{13})\)
- \(14\log_2(b) = \log_2(b^{14})\)
- \(2\log_2(c) = \log_2(c^2)\)
- Now the expression becomes: \[ \log_2(a^{13}) - \log_2(b^{14}) + \log_2(c^2) \]
- Next, apply the Quotient Rule: \[ \log_2\left(\frac{a^{13}}{b^{14}}\right) + \log_2(c^2) \]
- Finally, apply the Product Rule: \[ \log_2\left(\frac{a^{13} \cdot c^2}{b^{14}}\right) \]
- First, apply the Power Rule:
- This expression can indeed be rewritten as a single logarithm.
- This can be rewritten using the properties of logarithms:
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Expression: \(13\log_a \div 14\log_b \times 2\log_c\)
- This expression is ambiguous and lacks clarity due to the lack of complete operations (division and multiplication without specified arguments). However, based on typical notation, it's not possible to directly rewrite this as a single logarithm without knowing how the logs interact with each other. It’s best to clarify it further before determining if it can be condensed.
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Expression: \(13\log_2(a) - 14\log_3(b) + 2\log_5(c)\)
- Similar to the first case, the logarithms here are being taken to different bases (base 2, base 3, and base 5).
- Therefore, without converting all terms to a common logarithmic base, this expression cannot be condensed into a single logarithm.
Conclusion: Out of the three expressions, only the first expression \(13\log_2(a) - 14\log_2(b) + 2\log_2(c)\) can be rewritten as a single logarithm.