QuestionJuly 15, 2025

35. Condense the expression 3log_(7)v+6log_(7)w-(log_(7)u)/(3) to a single logarithm.

35. Condense the expression 3log_(7)v+6log_(7)w-(log_(7)u)/(3) to a single logarithm.
35. Condense the expression 3log_(7)v+6log_(7)w-(log_(7)u)/(3) to a single logarithm.

Solution
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Answer

\log_{7}\left(\frac{v^3 w^6}{u^{1/3}}\right) Explanation 1. Apply the Power Rule Use **a \cdot \log_b(x) = \log_b(x^a)** to rewrite each term: 3\log_{7}v = \log_{7}(v^3), 6\log_{7}w = \log_{7}(w^6), and \frac{\log_{7}u}{3} = \log_{7}(u^{1/3}). 2. Combine Using Addition/Subtraction Rules Use **\log_b(x) + \log_b(y) = \log_b(xy)** and **\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})** to combine: \log_{7}(v^3) + \log_{7}(w^6) - \log_{7}(u^{1/3}) = \log_{7}\left(\frac{v^3 w^6}{u^{1/3}}\right).

Explanation

1. Apply the Power Rule<br /> Use **$a \cdot \log_b(x) = \log_b(x^a)$** to rewrite each term: $3\log_{7}v = \log_{7}(v^3)$, $6\log_{7}w = \log_{7}(w^6)$, and $\frac{\log_{7}u}{3} = \log_{7}(u^{1/3})$.<br />2. Combine Using Addition/Subtraction Rules<br /> Use **$\log_b(x) + \log_b(y) = \log_b(xy)$** and **$\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})$** to combine: $\log_{7}(v^3) + \log_{7}(w^6) - \log_{7}(u^{1/3}) = \log_{7}\left(\frac{v^3 w^6}{u^{1/3}}\right)$.
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