Let's go through each of the questions step by step.
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Use the product rule to rewrite log(xy): The product rule for logarithms states that: \[ \log(xy) = \log(x) + \log(y) \] Therefore, the correct answer is log (x) + log (y).
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Use the quotient rule to rewrite log(x/y): The quotient rule for logarithms states that: \[ \log\left(\frac{x}{y}\right) = \log(x) - \log(y) \] Therefore, the correct answer is log (x) - log (y).
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Use the power rule to rewrite log(8): The power rule for logarithms states that: \[ \log(a^b) = b \cdot \log(a) \] Since \(8 = 2^3\), we have: \[ \log(8) = \log(2^3) = 3 \log(2) \] Therefore, the correct answer is 3 log (2).
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Identify the original base: If you're looking at \(\log_{b}(a) = c\) expressed in an image (or another context suggesting logarithmic bases), you'd typically identify the base from the context of an exponential equation. However, without seeing the image and based solely on common logarithmic context, possible bases for logarithms involve powers of integers. Common bases are often \(3, 10, \text{or } e\).
However, since three options were given (3, 27, 243), the exponents suggest:
- \(3^1 = 3\)
- \(3^3 = 27\)
- \(3^5 = 243\)
If we assume a consistent base (like \(3\)), then the original base may be 3 if the logarithm leads to those results.
If you have additional context or need specific checks related to the image you referred to, please provide that for more targeted insights.