Question

if x=logbc/loga,y=logca/logb,z=logab/logc,then the value of xyz-x-y-z=

Answer 1

logbc=logb +logc
logca=logc+log a
logab=loga +log b

Answer 2

We know that,

$\log_ab=\frac{\log b}{\log a}$

Using  this  property,

$x=\frac{\log bc}{\log a}=\log_abc\\\;\\y=\frac{\log ca}{\log b}=\log_bca\\\;\\z=\frac{\log ab}{\log c}=\log_cab$

Using  property:- $p=\log_qr\implies q^{p}=r$

$x=\log_abc\implies a^{x}=bc\\\;\\y=\log_bca\implies b^{x}=ca\\\;\\z=\log_cab\implies c^{x}=ab$

Now,

$a^{xyz}=(a^{x})^{yz}\\\;\\=(bc)^{yz}\\\;\\=(b^{y})^{z}\times(c^{z})^{y}\\\;\\=(ca)^{z}\times(ab)^{y}\\\;\\=c^{z}.a^{z}\times a^{y}.b^{y}\\\;\\=ab.a^{z+y}.b^{y}\\\;\\=a^{1+z+y}\times b.b^{y}\\\;\\=a^{1+z+y}\times b.ca\\\;\\=a^{1+z+y+1}\times bc\\\;\\=a^{2+z+y}.a^{x}\\\;\\a^{xyz}=a^{x+y+z+2}\\\;\\xyz=x+y+z+2\\\;\\xyz-x-y-z=2$