高中數(shù)學(xué)北師大版必修1課件第三章指數(shù)函數(shù)和對(duì)數(shù)函數(shù)24

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對(duì)數(shù)函數(shù)5.1

對(duì)數(shù)函數(shù)的概念1.掌握對(duì)數(shù)函數(shù)的概念.2.知道對(duì)數(shù)函數(shù)與指數(shù)函數(shù)互為反函數(shù).

1.對(duì)數(shù)函數(shù)(1)定義:一般地,我們把函數(shù)y=logax(a>0,a≠1)叫作對(duì)數(shù)函數(shù),a叫作對(duì)數(shù)函數(shù)的底數(shù),x是自變量,定義域是(0,+∞),值域是R.名師點(diǎn)撥1.由于指數(shù)函數(shù)y=ax中的底數(shù)a滿足a>0,a≠1,則對(duì)數(shù)函數(shù)y=logax中的底數(shù)a也必須滿足a>0,a≠1.2.對(duì)數(shù)函數(shù)的解析式同時(shí)滿足:對(duì)數(shù)符號(hào)前面的系數(shù)是1;對(duì)數(shù)的底數(shù)是不等于1的正實(shí)數(shù);對(duì)數(shù)的真數(shù)僅有自變量x.(2)兩類特殊的對(duì)數(shù)函數(shù).常用對(duì)數(shù)函數(shù):y=lgx,其底數(shù)為10.自然對(duì)數(shù)函數(shù):y=lnx,其底數(shù)為無理數(shù)e.【做一做1】

下列為對(duì)數(shù)函數(shù)的是(

)A.y=log1x

B.y=3log21xC.y=log19(x+1) D.y=log32x答案:D2.反函數(shù)對(duì)數(shù)函數(shù)y=logax(a>0,a≠1)和指數(shù)函數(shù)y=ax(a>0,a≠1)互為反函數(shù).【做一做2】

寫出下列函數(shù)的反函數(shù):3.互為反函數(shù)的兩個(gè)函數(shù)的關(guān)系(1)定義式:x與y互換,但通常用x表示自變量,y表示函數(shù).(2)圖像:關(guān)于直線y=x對(duì)稱.題型一題型二題型三題型一

判斷對(duì)數(shù)函數(shù)【例1】

下面是對(duì)數(shù)函數(shù)的是

(只填序號(hào)).

①y=log4x;②y=logx4;③y=log4(x+1);④y=log(-4)x.解析:根據(jù)對(duì)數(shù)函數(shù)的概念知,①函數(shù)y=log4x是對(duì)數(shù)函數(shù);②函數(shù)y=logx4的底數(shù)是自變量x,不是常數(shù),故不是對(duì)數(shù)函數(shù);③函數(shù)y=log4(x+1)的真數(shù)是x+1,不是自變量x,故不是對(duì)數(shù)函數(shù);④函數(shù)y=log(-4)x的底數(shù)是負(fù)數(shù),故不是對(duì)數(shù)函數(shù).答案:①反思判斷對(duì)數(shù)函數(shù)時(shí),要緊扣對(duì)數(shù)函數(shù)滿足的三個(gè)條件,缺一不可.題型一題型二題型三【變式訓(xùn)練1】

判斷下列函數(shù)是否為對(duì)數(shù)函數(shù):(1)y=2log3x;(2)y=log3(x-1).解:(1)中對(duì)數(shù)符號(hào)前面的系數(shù)是2,不是1,故不是對(duì)數(shù)函數(shù).(2)中真數(shù)為x-1,不是x,故不是對(duì)數(shù)函數(shù).題型一題型二題型三題型二

與對(duì)數(shù)函數(shù)有關(guān)的定義域問題【例2】

求下列函數(shù)的定義域:分析:要使函數(shù)有意義,則真數(shù)必須大于零,且分母不等于零.題型一題型二題型三反思求與對(duì)數(shù)有關(guān)的函數(shù)的定義域時(shí),除遵循前面已學(xué)過的求函數(shù)定義域的方法外,還要對(duì)這種函數(shù)自身有如下要求:(1)要特別注意真數(shù)大于零;(2)要注意對(duì)數(shù)的底數(shù);(3)按底數(shù)的取值應(yīng)用單調(diào)性,有針對(duì)性地解不等式.題型一題型二題型三【變式訓(xùn)練2】

求下列函數(shù)的定義域:題型一題型二題型三題型三

求反函數(shù)【例3】

寫出下列函數(shù)的反函數(shù):(1)y=log0.13x;

(2)y=3.05x.解:(1)y=log0.13x的反函數(shù)是y=0.13x.(2)y=3.05x的反函數(shù)是y=log3.05x.反思函數(shù)y=logax(a>0,a≠1)的反函數(shù)是y=ax(a>0,a≠1);函數(shù)y=ax(a>0,a≠1)的反函數(shù)是y=logax(a>0,a≠1).題型一題型二題型三【變式訓(xùn)練3】

求下列函數(shù)的反函數(shù):1234561已知函數(shù)f(x)=log2(x+1),若f(α)=1,則α=(

)A.0 B.1 C.2 D.3解析:∵f(α)=log2(α+1)=1,∴α+1=2.∴α=1.答案:B1234562已知函數(shù)y=log3x的定義域?yàn)?0,+∞),則它的反函數(shù)的值域是(

)A.(0,+∞) B.RC.(-∞,0) D.(0,1)解析:反函數(shù)的值域?yàn)樵瘮?shù)的定義域(0,+∞).答案:A123456答案:B1234564若函數(shù)y=f(x)的反函數(shù)圖像過點(diǎn)(1,5),則函數(shù)y=f(x)的圖像必過點(diǎn)(

)A.(1,1) B.(1,5) C.(5,1) D.(5,5)解析:原函數(shù)與它的反函數(shù)的圖像關(guān)于直線y=x對(duì)稱,點(diǎn)(1,5)關(guān)于直線y=x的對(duì)稱點(diǎn)為點(diǎn)(5,1),故選C.答案:C1234565函數(shù)y=lg(2x+1)的定義域是

.

1234566指出下列各組函數(shù)之間是否互為反函數(shù),并說明理由.(1)y=7x和y=log7x;解:(1),(3)組是,因?yàn)樗鼈?/p>