Question
Evaluate $: \frac{1}{\log _a b c+1}+\frac{1}{\log _b c a+1}+\frac{1}{\log _c a b+1}$
✓
Answer
$ \Rightarrow \frac{1}{\log _a b c+1}+\frac{1}{\log _b c a+1}+\frac{1}{\log _c a b+1}$
$ \Rightarrow \frac{1}{\log _a b c+\log _a a}+\frac{1}{\log _b c a+\log _b b}+\frac{1}{\log _c a b+\log _c c}$
$ \Rightarrow \frac{1}{\log _a a b c}+\frac{1}{\log _b a b c}+\frac{1}{\log _c a b c} \ldots\left[\because \log _{\mathrm{a}} \mathrm{b}+\log _{\mathrm{a}} \mathrm{c}=\log _{\mathrm{a}} \mathrm{bc}\right]$
$ \Rightarrow \frac{1}{\frac{\log a b c}{\log a}}+\frac{1}{\frac{\log a b c}{\log b}}+\frac{1}{\frac{\log a b c}{\log c}}$
$ \Rightarrow \frac{\log a+\log b+\log c}{\log a b c}$
$ \Rightarrow \frac{\log a b c}{\log a b c} \ldots \left[\because \log _{\mathrm{a}} \mathrm{b}+\log _{\mathrm{a}} \mathrm{c}=\log _{\mathrm{a}} \mathrm{bc}\right]$
$ \Rightarrow 1$
$ \Rightarrow \frac{1}{\log _a b c+\log _a a}+\frac{1}{\log _b c a+\log _b b}+\frac{1}{\log _c a b+\log _c c}$
$ \Rightarrow \frac{1}{\log _a a b c}+\frac{1}{\log _b a b c}+\frac{1}{\log _c a b c} \ldots\left[\because \log _{\mathrm{a}} \mathrm{b}+\log _{\mathrm{a}} \mathrm{c}=\log _{\mathrm{a}} \mathrm{bc}\right]$
$ \Rightarrow \frac{1}{\frac{\log a b c}{\log a}}+\frac{1}{\frac{\log a b c}{\log b}}+\frac{1}{\frac{\log a b c}{\log c}}$
$ \Rightarrow \frac{\log a+\log b+\log c}{\log a b c}$
$ \Rightarrow \frac{\log a b c}{\log a b c} \ldots \left[\because \log _{\mathrm{a}} \mathrm{b}+\log _{\mathrm{a}} \mathrm{c}=\log _{\mathrm{a}} \mathrm{bc}\right]$
$ \Rightarrow 1$
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