$\dfrac{1}{1+ \log_{u} \: vw} + \dfrac{1}{1+ \log_{v} \: wu} + \dfrac{1}{1+\log_{w} uv}$
$ = \dfrac{1}{\log_{u} \: u + \log_{u} \: vw} + \dfrac{1}{\log_{v} \: v + \log_{v} \: wu} + \dfrac{1}{\log_{w} \: w +\log_{w} \: uv}$
$ = \dfrac{1}{\log_{u} \: uvw} + \dfrac{1}{\log_{v} \: vwu} + \dfrac{1}{\log_{w} \: wuv}$
$ = \dfrac{1}{\log_{u} \: uvw} + \dfrac{1}{\log_{v} \: uvw} + \dfrac{1}{\log_{w}\: uvw}$
$ = \log_{uvw} \: u + \log_{uvw} \: v + \log_{uvw} \: w$
$ = \log_{uvw} \: uvw $
$ = 1\qquad\because\left(\log_{a} \: a = 1 \right)$
Hence $(C)$ is Correct.