HMMT 二月 2000 · GEN 赛 · 第 15 题
HMMT February 2000 — GEN Round — Problem 15
题目详情
- Whi h is greater: (3 ) or (5 ) ?
解析
- Noti e that 3 = 729 while 5 = 625, and sin e ln 5 > ln 3, it follo ws that 5 ln 3 < 3 ln 5, 3 5 3 5 5 3 so 5 5 ln 3 < 3 3 ln 5, and so b y la ws of logarithms, 5 ln 3 < 3 ln 5 . Again applying 3 5 5 (5 ) 3 (3 ) la ws of logarithms, it follo ws that ln (3 ) < ln (5 ) . So, sin e ln x is an in reasing 3 5 5 (5 ) (3 ) (3 ) 5 3 3 fun tion, it follo ws that (3 ) < (5 ) , and so it follo ws that (5 ) is greater . 4 6 (5 ) (3 ) One an also use the la ws of exp onen ts to redu e the v alues to 3 and 5 . The se ond is learly larger. x x x +2