Question

Sol. [Key Concept: A.P. condition among the listed numbers ⇒ derive the relations.]

$$3\log_{y}x = 3 + d$$

$$\begin{matrix} & 3\log_{z}y = 3 + 2d \\ & 7\log_{x}z = 3 + 3d \\ \Rightarrow & 3^{2} \times 7 = 3(1 + d)(3 + 2d)(3 + d) \\ \Rightarrow & \ (1 + d)(3 + d)(3 + 2d) = 21\ \Rightarrow \ d = \frac{1}{2} \\ \therefore\ & 6\log_{y}x = 7 \Rightarrow x^{6} = y^{7} \\ & 3\log_{z}y = 4 \Rightarrow y^{3} = z^{4} \\ & x^{18} = y^{21} = z^{28} \end{matrix}$$

✘ (a) $x^{6 =}y^{7} \neq \ y^{8}$

✔ (b) $y^{21} = z^{28} \Rightarrow \ y^{3 =}z^{4}$

✔ (c) $x^{6 =}y^{7}$

✔ (d) $x^{18} = z^{28} \Rightarrow \ x^{9 =}z^{14}$