Discussion
The value of the expression \(\dfrac{1}{1+\log_{u} vw} + \dfrac{1}{1+\log_{v} wu} + \dfrac{1}{1+\log_{w} uv}\) is _________.
Answer: C
The value of the expression is 1.
Change every logarithm to a common base: \(\log_{u} vw = \dfrac{\log v + \log w}{\log u}\). Then
\[ \begin{aligned} S &= \log u + \log v + \log w \\ 1 + \log_{u} vw &= \frac{S}{\log u} \\ \frac{1}{1 + \log_{u} vw} &= \frac{\log u}{S} \end{aligned} \]
The other two terms give \(\log v\) and \(\log w\) over the same denominator S, so the three add up to
\[ \frac{\log u + \log v + \log w}{S} = \frac{S}{S} = 1 \]
The result does not depend on u, v or w, so -1, 0 and 3 cannot be right.
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