HMMT 二月 2023 · 冲刺赛 · 第 5 题
HMMT February 2023 — Guts Round — Problem 5
题目详情
- [11] If a and b are positive real numbers such that a · 2 = 8 and a = 2, compute a 2 . { } 1
解析
- [11] If a and b are positive real numbers such that a · 2 = 8 and a = 2, compute a 2 . Proposed by: Daniel Hong Answer: 128 2 log a b 2 Solution: Taking log of both equations gives log a + b = 3 and b log a = 1. We wish to find a 2 ; 2 2 2 2 2 2 2 taking log of that gives (log a ) + b , which is equal to (log a + b ) − 2 b log a = 3 − 2 = 7. Hence, 2 2 2 2 7 our answer is 2 = 128. { } 1