An arithmetic progression of logarithms

Question

$1, \log_yx, \log_zy, -15\log_xz$ are in arithmetic progression, then which of the following are correct:

1. $z^3 = x$
2. $x = y^{-1}$
3. $z^{-3} = y$
4. $x = y^{-1} = z^3$

I tried converting the logs into a common base and then equate the difference between successive terms. However, this accomplished nothing.

Can you give me a hint on how to approach this problem?

Answer 1

All options are correct.

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, $$x = y^{1+d} = z^{(1+2d)(1+d)} = x^{\frac{-(1+d)(1+2d)(1+3d)}{15}}$$ $$\implies (1+d)(1+2d)(1+3d) = -15$$ $$\implies 6d^3 +11d^2+6d +16 = 0$$ $$\implies (d+2)(6d^2 - d+8) = 0$$ $$\implies d = -2$$

This leads us to: $$x^{1}=y^{-1}=z^{3}$$ This leads us options $(1),(2)$ and $(4)$ to be correct. Solving $y^{-1}=z^{3}$, we get $y=z^{-3}$ leading option $(3)$ to also being correct.

Hope this helps you.