HMMT 十一月 2023 · 冲刺赛 · 第 13 题

HMMT November 2023 — Guts Round — Problem 13

[9] Suppose x , y , and z are real numbers greater than 1 such that log z y x = 2 , log x z y = 4 , and log y x z = 8 . Compute log y . x

解析

[9] Suppose x , y , and z are real numbers greater than 1 such that log z y x = 2 , log x z y = 4 , and log y x z = 8 . Compute log y . x Proposed by: Rishabh Das √ Answer: 3 Solution: Taking log both sides of the first equation gives 2 log x log z = 1 2 y log x log z 2 2 = 1 . log y 2 Performing similar manipulations on other two equations, we get log x log z 2 2 = 1 log y 2 log y log x 2 2 = 2 log z 2 log z log y 2 2 = 3 . log x 2 √ 2 Multiplying the first and second equation gives (log x ) = 2 or log x = ± 2. Multiplying the second 2 2 √ 2 and third equation gives (log y ) = 6 or log y = ± 6. Thus, we have 2 2 √ √ log y 6 2 log y = = ± √ = ± 3 . x log x 2 2