To rewrite the expression \( \ln 7 + 3 \ln w - 2 \ln x \), we can apply the Product Rule, Quotient Rule, and Power Rule of Logarithms.
- Using the Power Rule:
- \( 3 \ln w \) can be rewritten as \( \ln (w^3) \).
- \( -2 \ln x \) can be rewritten as \( \ln (x^{-2}) \).
So the expression becomes: \[ \ln 7 + \ln (w^3) + \ln (x^{-2}). \]
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Using the Product Rule:
- Now we can combine the logarithms: \[ \ln (7 \cdot w^3) + \ln (x^{-2}) = \ln (7 \cdot w^3 \cdot x^{-2}). \]
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Continuing with Product Rule:
- Now, combining the last logarithm gives: \[ \ln \left(\frac{7 w^3}{x^2}\right). \]
Now we look at the provided options:
- Option #1: \( \ln(21 w^2 x) \) - This does not match our result.
- Option #2: \( \ln(7 w^3 x^2) \) - The exponent for \( x \) is negative, so this is incorrect.
- Option #3: \( \ln(7^3 w x) \) - This is incorrect because the exponents do not match.
None of the options match our derived expression \( \ln \left(\frac{7 w^3}{x^2}\right) \).
However, let's verify further. Upon revisiting, assuming all possible interpretations of exponents:
So we essentially get that the logarithmic representation does seem inconsistent since none matched outright, thus helping reaffirm our approach.
If forced to accept one as the "closest" or as the least incorrect under varied interpretation, option #2 might be the only dimensional equivalent despite the misalignment with x's power indicating why logarithmic accuracy matched would need precision acknowledgment.
Nevertheless, please affirm if there's need for tailored adjustments in interpretations or further parse among provided or broadly related options for clarity!