數(shù)學(xué)對(duì)數(shù)函數(shù)

函數(shù)、導(dǎo)數(shù)及其應(yīng)用2.7

對(duì)數(shù)函數(shù)運(yùn)算及其性質(zhì)對(duì)數(shù)的概念:如果ax=N(a>0,且a≠1),那么數(shù)x叫做以a為底N的對(duì)數(shù),記作x=logaN,其中a叫做對(duì)數(shù)的底數(shù),N叫做真數(shù).當(dāng)?shù)讛?shù)為10時(shí),稱為常用對(duì)數(shù),記作x=lgN;當(dāng)?shù)讛?shù)為無(wú)理數(shù)e時(shí),稱為自然對(duì)數(shù),記作x=ln

N.運(yùn)算法則:如果a>0,且a≠1,N>0,M>0,有①loga(MN)=logaM+logaN;m③lo

M

=alog

M.logaN=b;②loga

M

=logaM-logaN;Ng

namn3.常用公式:①指對(duì)互化公式ab=N②對(duì)數(shù)恒等式aloga

N

=N;(m>0,且m≠1).③換底公式logaM=logm

Mlogm

a無(wú)法顯示該圖片。y=logaxa>10<a<1圖象二、對(duì)數(shù)函數(shù)的圖象及性質(zhì)三、指數(shù)函數(shù)與對(duì)數(shù)函數(shù)的關(guān)系指數(shù)函數(shù)y=ax與對(duì)數(shù)函數(shù)y=logax互為反函數(shù),它們的圖象關(guān)于直線y=x對(duì)稱.性質(zhì)定義域:(0,+∞)值域:R奇偶性:非奇非偶函數(shù)過定點(diǎn):(1,0)當(dāng)x∈(0,1)時(shí),y<0,當(dāng)x∈(1,+∞)時(shí),y>0當(dāng)x∈(0,1)時(shí),y>0,當(dāng)x∈(1,+∞)時(shí),y<0在(0,+∞)是增函數(shù)在(0,+∞)是減函數(shù)無(wú)法顯示該圖片。無(wú)法顯示該圖片。無(wú)法顯示該圖片。無(wú)法顯示該圖片。1、對(duì)數(shù)計(jì)算2、已知a＝log23.6，b＝log43.2，c＝log43.6，則(

)A．a(chǎn)>b>cC．b>a>cB．a(chǎn)>c>bD．c>a>b3、(2013·新課標(biāo)全國(guó)Ⅱ理)設(shè)a＝log36，b＝log510，c＝log714，則(

)A．c>b>aC．a(chǎn)>c>bB．b>c>aD．a(chǎn)>b>c4、已知函數(shù)f(x)＝log2(x2－2ax＋3)．(1)若a＝1，求f(x)的值域；(2)若a＝2，求函數(shù)f(x)的定義域及單調(diào)區(qū)間；(3)若函數(shù)f(x)的定義域?yàn)镽，求實(shí)數(shù)a的取值范圍；(4)若函數(shù)f(x)的值域?yàn)镽，求實(shí)數(shù)a的取值范圍(5)若函數(shù)f(x)在[2，＋∞)上單調(diào)遞增，求實(shí)數(shù)a的取值范圍；(6)若函數(shù)f(x)在[－1，＋∞)上有意義，求實(shí)數(shù)a的取值范圍．無(wú)法顯示該圖片。6、已知函數(shù)f(x)=log4(4x+1)+kx(k∈R)是偶函數(shù).求k的值;解不等式f(x)>1.7、(能力綜合)若f(x)=x2-x+b,且f(log2a)=b,log2f(a)=2(a≠1).求f