Question

Determine the relations in x, y and z if $1,lo{g}_{y}x,lo{g}_{z}y,-15lo{g}_{x}z$ are in A.P.

Answer 1

If d be the common difference of A.P., then
$lo{g}_{y}x=1+d,lo{g}_{z}y=1+2d,-15lo{g}_{x}z=1+3d$
$\therefore x=y^{1+d},y=z^{1+2d},z=x^{-\frac{1+3d}{15}}$
Eliminating y and z, we get
$x=z^{(1+d)(1+2d)}=x^{-\frac{(1-3d)(1+d)(1+2d)}{15}}$
$\therefore (1+d)(1+2d)(1+3d)=-15$
or $6d^3+11d^2+6d+16=0$
$(d+2)(6d^2-d+8)=0$
$\therefore d=-2$
The other factor gives imaginary values of d.
$\therefore x=y^{-1},y=z^{-3},z=x^{1/3}$
or $x=y^{-1}=z^3,y=z^{-3}$.