Question

If $\log x:\log y:\log z=(y-z):(z-x):(x-y)$, then which of the following condition is true?

• A. $x^y.y^z.z^x=1$
• B. $x^x.y^y.z^z=1$
• C. $\sqrt[x]{xx}.\sqrt[y]{yy}.\sqrt[z]{zz}.=1$
• D. none of these

Answer 1

The correct option is B $x^x.y^y.z^z=1$

Let $\frac{\log x}{y-z}=\frac{\log y}{z-x}=\frac{\log z}{y-z}=k$

Now,

$x\log x+y\log y+z\log z=k[x(y-z)+y(z-x)+z(y-z)]$

$\Rightarrow \log x^x+\log y^y+\log z^z=0$

$\Rightarrow \log x^x{y}^y{z}^z=0$

$\Rightarrow x^x{y}^y{z}^z=1$

Thus the correct answer is option $B$