Without expanding prove that: logxxyzlogxylogxzlogyxyz1logyzlogzxyzlogzy1=0 - Maths - Differential Equations

Question

Without expanding prove that:
log x x y z log x y log x z log y x y z 1 log y z log z x y z log z
y 1 = 0

Neha Sethi · Accepted Answer

Dear student
We havelogxxyzlogxylogxzlogyxyz1logyzlogzxyzlogzy1=logxyzlogxlogylogxlogzlogxlogxyzlogylogylogylogzlogylogxyzlogzlogylogzlogzlogz   ∵logab=logblogaTaking 1logx, 1logy and 1logz common from R1 ,R2 and R3 respectively
=1logx logy logzlogxyzlogylogzlogxyzlogylogzlogxyzlogylogz=0×1logx logy logz      ∵R1,R2 and R3 are identical
=0
Regards

Without expanding prove that: log x x y z log x y log x z log y x y z 1 log y z log z x y z log z y 1 = 0

Without expanding prove that:
$|\begin{matrix} \log_{x} x y z & \log_{x} y & \log_{x} z \\ \log_{y} x y z & 1 & \log_{y} z \\ \log_{z} x y z & \log_{z} y & 1 \end{matrix}| = 0$