Question

比較下列各組中兩個(gè)值的大小.

(1)log₃1.9,log₃2;             (2)log_0.90.1,log_0.92;

(3)log₃5,log₅3;                (4)log₂3,log₀.32;

(5)log_aπ,log_a3.141.

Answer 1

思路分析:比較兩個(gè)對(duì)數(shù)值的大小:同底可利用對(duì)數(shù)函數(shù)的單調(diào)性,如(1)(2);若底數(shù)、真數(shù)都不同可以借助常數(shù)(常用的-1,0,1)為媒介間接比較大小,如(3)(4);若真數(shù)相同,底數(shù)不同可以借助對(duì)數(shù)函數(shù)圖象來(lái)比較大小;若底數(shù)與1的大小關(guān)系不確定,要分情況討論.

解:(1)因?yàn)閥=log₃x在(0,+∞)上是增函數(shù),所以log₃1.9＜log₃2.

(2)因?yàn)閥=log_0.9x在(0,+∞)上是減函數(shù),所以log_0.90.1＞log_0.92.

(3)因?yàn)閘og₃5＞log₃3=1=log₅5＞log₅3,所以log₃5＞log₅3.

(4)因?yàn)閘og₂3＞log₂2=1,log_0.32＜0,所以log₂3＞log₀.₃2.

(5)當(dāng)a＞1時(shí),log_aπ＞log_a3.141;

當(dāng)0＜a＜1時(shí),log_aπ＜log_a3.141.