Question

Prove that $x^{logy-logz}\times y^{logz-logx}\times z^{logx-logy}=1$

Answer 1

Let $LHS=x^{\log y-\log z}\times y^{\log z-\log x}\times z^{\log x-\log y}$

$=e^{\log (x^{\log y-\log z}\times y^{\log z-\log x}\times z^{\log x-\log y)}}$ $(\because e^{\log a}=a)$

$=e^{\log x^{(\log y-\log z)}+\log y^{(\log z-\log x)}+\log z^{(\log x-\log y)}}$

$=e^{(\log y-\log z)\log x+(\log z-\log x)\log y+(\log x-\log y)\log z}$

$(\because \log a^b=b\log a)$

$=e^{\log y.\log x-\log z+\log x+\log z\log y-\log x\log y\log x\log z-\log y\log z}$

$=e^o$

$=1$

$=RHS$