If $(1), (log_yx), (log_zy), (-15log_xz)$ are in AP then...
(A)$z^3=x$
(B)$x=y^{-1}$
(C)$z^{-3}=y$
(D)$x=y^{-1}=z^3$
(Multiple answers may be correct)
I tried many approaches, one of them being:
$1+log_zy=2log_yx$
$log_zz+log_zy=log_yx^2$
$log_z(yz)=log_yx^2$
But the fact that the bases are different is causing the main problem as I don't know how to proceed further. It would be great if anybody could give me a hint for solving such questions.
Let $d$ be the common difference of the $A.P.$
Then,
$$log_yx = 1 +d \implies x = y^{1+d}$$ $$log_zy = 1+2d \implies y = z^{1+2d}$$ $$-15log_xz = 1+3d \implies z = x^{\frac{-(1+3d)}{15}}$$ Hence, $$x = y^{1+d} = z^{(1+2d)(1+d)} = x^{\frac{-(1+d)(1+2d)(1+3d)}{15}}$$ $$\implies (1+d)(1+2d)(1+3d) = -15$$ $$\implies 6d^3 +11d^2+6d +16 = 0$$ $$\implies (d+2)(6d^2 - d+8) = 0$$ $$\implies d = -2$$
$$\implies x = y^{-1} = z^{3}$$