Symmetric Logarithms

Algebra Level 3

Given that $\frac { \log x }{y-z} = \frac { \log y }{z-x} = \frac { \log z }{x-y}$ . What is the value of $x^x \cdot y^y \cdot z^z$ ?

\log { \left ( { x }^{ x } { y }^{ y } { z }^{ z } \right ) }

1

x+y+z

1 solution

Ronak Agarwal
Aug 22, 2014

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

Hence we have : $log(x)=(y-z)*k$

$log(y)=(z-x)*k$

$log(z)=(x-y)*k$

So we write :

$xlog(x)+ylog(y)+zlog(z)=x(y-z)k+y(z-x)k+z(x-y)k=0$

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

Finally ${x}^{x}{y}^{y}{z}^{z}=1$

Just the way you do it

Satyajit Ghosh - 6 years, 9 months ago

Yes Same way. A fifteen booklet question:p

Kushagra Sahni - 6 years, 9 months ago

I mean Fiitjee

Kushagra Sahni - 6 years, 9 months ago