Question

Suppose $x$,$y$,$z$ $>0$ and are not equal to $1$ and $\log x+\log y+\log z=0$. What is the value of $x^{\frac{1}{\log y}+\frac{1}{\log z}}\times y^{\frac{1}{\log z}+\frac{1}{\log x}}\times z^{\frac{1}{\log x}+\frac{1}{\log y}}$.
(All log functions have base $10$)

• A. ${10}^{-3}$
• B. ${10}^{-4}$
• C. ${10}^{-2}$
• D. none of these

Answer 1

The correct option is A ${10}^{-3}$

Let, $K=x^{\frac{1}{\log y}+\frac{1}{\log z}}\times y^{\frac{1}{\log z}+\frac{1}{\log x}}\times z^{\frac{1}{\log x}+\frac{1}{\log y}}$

Taking $\log$ on both the sides,
$\log K=\log x[\frac{1}{\log y}+\frac{1}{\log z}]+\log y[\frac{1}{\log z}+\frac{1}{\log x}]+\log z[\frac{1}{\log x}+\frac{1}{\log y}]\to \left(i\right)$

Putting $\log x+\log y+\log z=0$ (given), we get

$\frac{\log x}{\log y}+\frac{\log z}{\log y}=-1;\frac{\log y}{\log x}+\frac{\log z}{\log x}=-1;\frac{\log x}{\log z}+\frac{\log y}{\log z}=-1$

$\therefore$ R.H.S. of Eq.(i)$=-3$
$\Rightarrow {\log }_{10}K=-3$ or $K={10}^{-3}$