Question

If $x,y,z$ are positive then the minimum value of $x^{logy-logz}+y^{logz-logx}+z^{logx-logy}$ is

• A. $3$
• B. $1$
• C. $9$
• D. $16$

Answer 1

The correct option is A $3$

Let

$a=x^{\log y-\log x}$

$b=y^{\log z-\log x}$

$c=z^{\log x-\log y}$

Now,

$\log \left(abc\right)=\log a+\log b+\log c$

$\Rightarrow \log \left(abc\right)=\log x^{\log y-\log x}+\log y^{\log z-\log x}+\log z^{\log x-\log y}$

$\Rightarrow \log \left(abc\right)=(\log y-\log x)\log x+(\log z-\log x)\log y+(\log x-\log y)\log z$

$\Rightarrow \log \left(abc\right)=0$

$\Rightarrow abc=1$

As a,b,c are positive,

$\therefore AM\ge GM$

$\Rightarrow \frac{a+b+c}{3}\ge (abc{)}^{\frac{1}{3}}$

$\Rightarrow \frac{a+b+c}{3}\ge (1{)}^{\frac{1}{3}}$

$\Rightarrow a+b+c\ge 3$