Question 68

Simplify : $$\left[\frac{1}{\log_{xy} (xyz)} + \frac{1}{\log_{yz} (xyz)} + \frac{1}{\log_{zx} (xyz)}\right]$$

Solution

Expression : $$\left[\frac{1}{\log_{xy} (xyz)} + \frac{1}{\log_{yz} (xyz)} + \frac{1}{\log_{zx} (xyz)}\right]$$

Using, $$\log_m(n)=\frac{1}{\log_n(m)}$$ and $$\log(mn)=\log(m)+\log(n)$$

= $$\log_{xyz}(xy)+\log_{xyz}(yz)+\log_{xyz}(zx)$$

= $$\log_{xyz}(xy\times yz\times zx)$$

= $$\log_{xyz}(xyz)^2$$

= $$2\log_{xyz}(xyz)$$ [$$log_m(m)=1$$]

= $$2\times1=2$$

=> Ans - (D)

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