Question

If x⁴² = y²⁷= z³⁶ such that x ≠ 1, y ≠ 1, z ≠ 1, then 16/3 log_yx, log_zy and 4/9 log_xz are in

विकल्प

• Arithmetic progression

• Geometric progression

• Harmonic progression

• None of these

Answer 1

Let x⁴² = y²⁷= z³⁶ = k
⇒ 42 log x= log k, 27 log y = log k, 36 log z = log k

⇒ `(42 logx)/(27 log y) = 1 ⇒ 14/9 log_y x = 1`

⇒ `log_y x = 9/14 ⇒ 16/3 log_y x = 144/42`

Similarly `log_x y = 4/3` and
`log_x z = 7/6 ⇒ 4/9 log_x z = 28/54`

`(log_z  y)^2 = 16/3 log_y x xx 4/9 log_x z`

`(4/3)^2 = (144/42)(28/54) = 16/9`

Hence, the three terms are in Geometric progression.