對(duì)數(shù)函數(shù)的概念

對(duì)數(shù)函數(shù)的概念復(fù)習(xí)回顧換底公式常用結(jié)論在細(xì)胞分裂的問題中，細(xì)胞分裂個(gè)數(shù)y和分裂次數(shù)x的函數(shù)關(guān)系,用正整數(shù)指數(shù)函數(shù)y=2x表示.在學(xué)習(xí)過程中我們已經(jīng)反它推廣到實(shí)數(shù)指數(shù)函數(shù).分裂?次細(xì)胞個(gè)數(shù)1萬10萬在y=2x中知y求xx=log2y一般的指數(shù)函數(shù)y=ax(a>0,a≠1)中的兩個(gè)變量,能不能把y當(dāng)作自變量,使得x是y的函數(shù)?對(duì)于任意y∈(0,+∞)有唯一x∈R滿足y=ax把y當(dāng)作自變量,x是y的函數(shù)x=logay(a>0,a≠1)x1x2y2y1xyy=ax(a>1)x1≠x2y1≠y2y=ax(a>0,a≠1),對(duì)于x每一個(gè)確定值,y都有唯一確定的值和它對(duì)應(yīng).R{y|y>0}一一對(duì)應(yīng)R{y|y>0}x=logay對(duì)數(shù)函數(shù)y>0a>0,a≠1把函數(shù)y=logax(a>0,a≠1)叫作對(duì)數(shù)函數(shù)a為對(duì)數(shù)函數(shù)的底數(shù)10為底的對(duì)數(shù)函數(shù)y=lgx為常用對(duì)數(shù)函數(shù)以無理數(shù)e為底的對(duì)數(shù)函數(shù)y=lnx為自然對(duì)數(shù)函數(shù)例1計(jì)算：(1)計(jì)算對(duì)數(shù)函數(shù)y=log2x對(duì)應(yīng)于x取1,2,4時(shí)的函數(shù)值；(2)計(jì)算常用對(duì)數(shù)函數(shù)y=lgx對(duì)應(yīng)于x取1,10,100,0.1時(shí)的函數(shù)值.解(1)當(dāng)x=1時(shí),y=log2x=log21=0,

當(dāng)x=2時(shí),y=log2x=log22=1,

當(dāng)x=4時(shí),y=log2x=log24=2;(2)當(dāng)x=1時(shí),y=lgx=lg1=0,

當(dāng)x=10時(shí),y=lgx=lg10=1,

當(dāng)x=100時(shí),y=lgx=lg100=2,

當(dāng)x=0.1時(shí),y=lgx=lg0.1=-1.指數(shù)函數(shù)y=ax與對(duì)數(shù)函數(shù)x=logay(a>0,a≠1)有什么關(guān)系?函數(shù)自變量因變量定義域值域y=axxyR(0,+∞)x=logayyx(0,+∞)R稱這兩個(gè)函數(shù)互為反函數(shù)對(duì)應(yīng)關(guān)系互逆指數(shù)函數(shù)y=ax是對(duì)數(shù)函數(shù)x=logay(a>0,a≠1)的反函數(shù)指數(shù)函數(shù)y=ax(a>0,a≠1)對(duì)數(shù)函數(shù)y=logax(a>0,a≠1)反函數(shù)例2寫出下列對(duì)數(shù)函數(shù)的反函數(shù):(1)y=lgx;

解(1)對(duì)數(shù)函數(shù)y=lgx,它的底數(shù)是它的反函數(shù)是指數(shù)函數(shù)10y=10x(2)對(duì)數(shù)函數(shù)它的底數(shù)是它的反函數(shù)是指數(shù)函數(shù)例3寫出下列指數(shù)函數(shù)的反函數(shù):(1)y=5x

解(1)指數(shù)函數(shù)y=5x,它的底數(shù)是5它的反函數(shù)是對(duì)數(shù)函數(shù)y=log5x;(2)指數(shù)函數(shù),它的底數(shù)是,它的反函數(shù)是對(duì)數(shù)函數(shù)

練習(xí)1.計(jì)算:(1)計(jì)算對(duì)數(shù)函數(shù)

對(duì)應(yīng)于x取0.25,0.5,1,2,4,8時(shí)的函數(shù)值;

(2)計(jì)算常用對(duì)數(shù)函數(shù)y=lgx對(duì)應(yīng)于x取0.1,0.0001,1,100時(shí)的函數(shù)值.練習(xí)2.說出下列各組函數(shù)之間的關(guān)系:(1)y=10x和y=lgx；(2)y=2x和y=log2x;(3)y=ex和y=lnx.互為反函數(shù),定義域和值域互換,對(duì)應(yīng)法則互逆練習(xí)3.寫出下列對(duì)數(shù)函數(shù)的反函數(shù):(1)y=log2.5x;(2)y=logπx;4.寫出下列指數(shù)函數(shù)的反函數(shù):(1)y=4x;

(2)y=1.4x;(1)y=2.5x(2)y=πx(1)y=log4x(2)y=log1.4x小結(jié)對(duì)數(shù)函數(shù)的概念反函數(shù)定義域和值