高中數(shù)學(xué)第二章2.2對(duì)數(shù)函數(shù)2.2.2對(duì)數(shù)函數(shù)及其性質(zhì)（一）學(xué)案（含解析）新人教A版.docx

2.2.2對(duì)數(shù)函數(shù)及其性質(zhì)(一)學(xué)習(xí)目標(biāo)1.理解對(duì)數(shù)函數(shù)的概念.2.掌握對(duì)數(shù)函數(shù)的性質(zhì).3.了解對(duì)數(shù)函數(shù)在生產(chǎn)實(shí)際中的簡(jiǎn)單應(yīng)用知識(shí)點(diǎn)一對(duì)數(shù)函數(shù)的概念思考已知函數(shù)y2x，那么反過來，x是否為關(guān)于y的函數(shù)？答案由于y2x是單調(diào)函數(shù)，所以對(duì)于任意y(0，)都有唯一確定的x與之對(duì)應(yīng)，故x也是關(guān)于y的函數(shù)，其函數(shù)關(guān)系式是xlog2y，此處y(0，)習(xí)慣上用x，y分別表示自變量、因變量上式可改為ylog2x，x(0，)梳理一般地，把函數(shù)ylogax(a0，且a1)叫做對(duì)數(shù)函數(shù)，其中x是自變量，函數(shù)的定義域是(0，)知識(shí)點(diǎn)二對(duì)數(shù)函數(shù)的圖象與性質(zhì)對(duì)數(shù)函數(shù)ylogax(a0，且a1)的圖象和性質(zhì)如下表：定義ylogax (a0，且a1)底數(shù)a10a0.()2y2log2x是對(duì)數(shù)函數(shù)()3yax與ylogax的單調(diào)區(qū)間相同()4由loga10，可得ylogax恒過定點(diǎn)(1,0)()類型一對(duì)數(shù)函數(shù)的定義域的應(yīng)用例1求下列函數(shù)的定義域(1)yloga(3x)loga(3x)；(2)ylog2(164x)考點(diǎn)對(duì)數(shù)函數(shù)的定義域題點(diǎn)對(duì)數(shù)函數(shù)的定義域解(1)由得3x3，函數(shù)的定義域是x|3x0，得4x1642，由指數(shù)函數(shù)的單調(diào)性得x2，函數(shù)ylog2(164x)的定義域?yàn)閤|x3.函數(shù)yloga(x3)loga(x3)的定義域?yàn)閤|x32求函數(shù)yloga(x3)(x3)的定義域，相比引申探究1，定義域有何變化？解(x3)(x3)0，即或解得x3.函數(shù)yloga(x3)(x3)的定義域?yàn)閤|x3相比引申探究1，函數(shù)yloga(x3)(x3)的定義域多了(，3)這個(gè)區(qū)間，原因是對(duì)于yloga(x3)(x3)，要使對(duì)數(shù)有意義，只需(x3)與(x3)同號(hào)，而對(duì)于yloga(x3)loga(x3)，要使對(duì)數(shù)有意義，必須(x3)與(x3)同時(shí)大于0.反思與感悟求含對(duì)數(shù)式的函數(shù)定義域關(guān)鍵是真數(shù)大于0，底數(shù)大于0且不為1.如需對(duì)函數(shù)式變形，需注意真數(shù)底數(shù)的取值范圍是否改變跟蹤訓(xùn)練1求下列函數(shù)的定義域(1)y；(2)ylog(x1)(164x)；考點(diǎn)對(duì)數(shù)函數(shù)的定義域題點(diǎn)對(duì)數(shù)函數(shù)的定義域解(1)要使函數(shù)有意義，需即即3x2或x2，故所求函數(shù)的定義域?yàn)?3，2)2，)(2)要使函數(shù)有意義，需即所以1x2，且x0，故所求函數(shù)的定義域?yàn)閤|1x0，且a1)考點(diǎn)對(duì)數(shù)值大小比較題點(diǎn)對(duì)數(shù)值大小比較解(1)考察對(duì)數(shù)函數(shù)ylog2x，因?yàn)樗牡讛?shù)21，所以它在(0，)上是增函數(shù)，又3.48.5，于是log23.4log28.5.(2)考察對(duì)數(shù)函數(shù)ylog0.3x，因?yàn)樗牡讛?shù)00.3log0.32.7.(3)當(dāng)a1時(shí)，ylogax在(0，)上是增函數(shù)，又5.15.9，于是loga5.1loga5.9；當(dāng)0aloga5.9.綜上，當(dāng)a1時(shí)，loga5.1loga5.9，當(dāng)0a1時(shí)，loga5.1loga5.9.反思與感悟比較兩個(gè)同底數(shù)的對(duì)數(shù)大小，首先要根據(jù)對(duì)數(shù)底數(shù)來判斷對(duì)數(shù)函數(shù)的增減性；然后比較真數(shù)大小，再利用對(duì)數(shù)函數(shù)的增減性判斷兩對(duì)數(shù)值的大小對(duì)于底數(shù)以字母形式出現(xiàn)的，需要對(duì)底數(shù)a進(jìn)行討論對(duì)于不同底的對(duì)數(shù)，可以估算范圍，如log22log23log24，即1log232，從而借助中間值比較大小跟蹤訓(xùn)練2設(shè)alog3，blog2，clog3，則()AabcBacbCbacDbca考點(diǎn)對(duì)數(shù)值大小比較題點(diǎn)對(duì)數(shù)值大小比較答案A解析alog31，blog23，其中l(wèi)og22log230，3x11.ylog2x在(0，)上單調(diào)遞增，log2(3x1)log210.即f(x)的值域?yàn)?0，)反思與感悟在函數(shù)三要素中，值域從屬于定義域和對(duì)應(yīng)關(guān)系故求ylogaf(x)型函數(shù)的值域必先求定義域，進(jìn)而確定f(x)的范圍，再利用對(duì)數(shù)函數(shù)ylogax的單調(diào)性求出logaf(x)的取值范圍跟蹤訓(xùn)練3已知f(x)log2(1x)log2(x3)，求f(x)的定義域、值城考點(diǎn)對(duì)數(shù)函數(shù)的值域題點(diǎn)真數(shù)為二次函數(shù)的對(duì)數(shù)型函數(shù)的值域解要使函數(shù)式有意義，需解得定義域?yàn)?3,1)f(x)log2(1x)(x3)log2(x1)24x(3,1)，(x1)24(0,4log2(x1)24(，2即f(x)的值域?yàn)?，2類型三對(duì)數(shù)函數(shù)的圖象例4畫出函數(shù)ylg|x1|的圖象考點(diǎn)對(duì)數(shù)函數(shù)的圖象題點(diǎn)含絕對(duì)值的對(duì)數(shù)函數(shù)的圖象解(1)先畫出函數(shù)ylgx的圖象(如圖)(2)再畫出函數(shù)ylg|x|的圖象(如圖)(3)最后畫出函數(shù)ylg|x1|的圖象(如圖)反思與感悟現(xiàn)在畫圖象很少單純依靠描點(diǎn)，大多是以基本初等函數(shù)為原料加工，所以一方面要掌握一些常見的平移、對(duì)稱變換的結(jié)論，另一方面要關(guān)注定義域、值域、單調(diào)性、關(guān)鍵點(diǎn)跟蹤訓(xùn)練4畫出函數(shù)y|lg(x1)|的圖象考點(diǎn)對(duì)數(shù)函數(shù)的圖象題點(diǎn)含絕對(duì)值的對(duì)數(shù)函數(shù)的圖象解(1)先畫出函數(shù)ylgx的圖象(如圖)(2)再畫出函數(shù)ylg(x1)的圖象(如圖)(3)再畫出函數(shù)y|lg(x1)|的圖象(如圖)1下列函數(shù)為對(duì)數(shù)函數(shù)的是()Aylogax1(a0且a1)Byloga(2x)(a0且a1)Cylog(a1)x(a1且a2)Dy2logax(a0且a1)考點(diǎn)對(duì)數(shù)函數(shù)的概念題點(diǎn)對(duì)數(shù)函數(shù)的概念答案C2函數(shù)ylog2(x2)的定義域是()A(0，) B(1，)C(2，) D4，)考點(diǎn)對(duì)數(shù)函數(shù)的定義域題點(diǎn)對(duì)數(shù)函數(shù)的定義域答案C3函數(shù)y2log4(1x)的圖象大致是()考點(diǎn)對(duì)數(shù)函數(shù)的圖象題點(diǎn)對(duì)數(shù)函數(shù)的圖象答案C解析函數(shù)y2log4(1x)的定義域?yàn)?，1)，排除A，B；又函數(shù)y2log4(1x)在定義域內(nèi)單調(diào)遞減，排除D.故選C.4函數(shù)f(x)log0.2(2x1)的值域?yàn)開考點(diǎn)對(duì)數(shù)函數(shù)的值域題點(diǎn)對(duì)數(shù)函數(shù)的值域答案(，0)5若函數(shù)f(x)2loga(2x)3(a0，且a1)過定點(diǎn)P，則點(diǎn)P的坐標(biāo)是_考點(diǎn)對(duì)數(shù)函數(shù)的性質(zhì)題點(diǎn)對(duì)數(shù)函數(shù)圖象過定點(diǎn)問題答案(1,3)1含有對(duì)數(shù)符號(hào)“l(fā)og”的函數(shù)不一定是對(duì)數(shù)函數(shù)判斷一個(gè)函數(shù)是否為對(duì)數(shù)函數(shù)，不僅要含有對(duì)數(shù)符號(hào)“l(fā)og”，還要符合對(duì)數(shù)函數(shù)的概念，即形如ylogax(a0，且a1)的形式如：y2log2x，ylog5都不是對(duì)數(shù)函數(shù)，可稱其為對(duì)數(shù)型函數(shù)2研究ylogaf(x)的性質(zhì)如定義域、值域、比較大小，均需依托對(duì)數(shù)函數(shù)的相應(yīng)性質(zhì)一、選擇題1給出下列函數(shù)：ylogx2；ylog3(x1)；ylog(x1)x；ylogx.其中是對(duì)數(shù)函數(shù)的有()A1個(gè)B2個(gè)C3個(gè)D4個(gè)考點(diǎn)對(duì)數(shù)函數(shù)的概念題點(diǎn)對(duì)數(shù)函數(shù)的概念答案A解析不是對(duì)數(shù)函數(shù)，因?yàn)閷?duì)數(shù)的真數(shù)不是只含有自變量x；不是對(duì)數(shù)函數(shù)，因?yàn)閷?duì)數(shù)的底數(shù)不是常數(shù)；是對(duì)數(shù)函數(shù)2已知函數(shù)f(x)的定義域?yàn)镸，g(x)ln(1x)的定義域?yàn)镹，則MN等于()Ax|x1Bx|x1Cx|1x0x|x0x|x1，MNx|1x0，且a1，函數(shù)yax與yloga(x)的圖象只能是下圖中的()考點(diǎn)對(duì)數(shù)函數(shù)的圖象題點(diǎn)同一坐標(biāo)系下的指數(shù)函數(shù)與對(duì)數(shù)函數(shù)的圖象答案B解析yax與yloga(x)的單調(diào)性相反，排除A，D.yloga(x)的定義域?yàn)?，0)，排除C，故選B.4已知函數(shù)f(x)loga(x2)，若圖象過點(diǎn)(6,3)，則f(2)的值為()A2B2C.D考點(diǎn)對(duì)數(shù)函數(shù)的性質(zhì)題點(diǎn)對(duì)數(shù)函數(shù)圖象過定點(diǎn)問題答案B解析代入(6,3)，3loga(62)loga8，即a38，a2.f(x)log2(x2)，f(2)log2(22)2.5若函數(shù)f(x)loga(xb)的圖象如圖所示：其中a，b為常數(shù)，則函數(shù)g(x)axb的圖象大致是()考點(diǎn)對(duì)數(shù)函數(shù)的圖象題點(diǎn)同一坐標(biāo)系下的指數(shù)函數(shù)與對(duì)數(shù)函數(shù)的圖象答案D解析由f(x)的圖象可知0a1,0blog0.52.3Blog34log65Clog34log56Dlogeln考點(diǎn)對(duì)數(shù)值大小比較題點(diǎn)對(duì)數(shù)值大小比較答案D解析對(duì)A，根據(jù)ylog0.5x為單調(diào)減函數(shù)易知正確對(duì)B，由log34log331log55log65可知正確對(duì)C，由log341log31log31log5log56可知正確對(duì)D，由e1，得ln1loge可知錯(cuò)誤7已知f(x)2log3x，x，則f(x)的最小值為()A2B3C4D0考點(diǎn)對(duì)數(shù)函數(shù)的值域題點(diǎn)對(duì)數(shù)函數(shù)的值域答案A解析x9，log3log3xlog39，即4log3x2，22log3x4.當(dāng)x時(shí)，f(x)min2.8已知函數(shù)f(x)loga|x1|在(1,0)上有f(x)0，那么()Af(x)在(，0)上是增函數(shù)Bf(x)在(，0)上是減函數(shù)Cf(x)在(，1)上是增函數(shù)Df(x)在(，1)上是減函數(shù)考點(diǎn)對(duì)數(shù)函數(shù)的圖象題點(diǎn)含絕對(duì)值的對(duì)數(shù)函數(shù)的圖象答案C解析當(dāng)x(1,0)時(shí)，|x1|(0,1)，loga|x1|0，0a1，畫出f(x)的圖象如圖：由圖可知選C.二、填空題9.已知函數(shù)f(x)的圖象如圖所示，則函數(shù)g(x)logf(x)的定義域是_考點(diǎn)對(duì)數(shù)函數(shù)的定義域題點(diǎn)對(duì)數(shù)函數(shù)的定義域答案x|20，由所給圖象可知f(x)0的解集為x|2cb解析因?yàn)?，所以alog21，所以blog1，所以021，即0ccb.11已知函數(shù)f(x)|lgx|，若0ab，且f(a)f(b)，則a4b的取值范圍是_考點(diǎn)對(duì)數(shù)函數(shù)的圖象題點(diǎn)含絕對(duì)值的對(duì)數(shù)函數(shù)的圖象答案(5，)解析因?yàn)閒(a)f(b)，且0ab，所以0a1g(1)15，即a4b的取值范圍是(5，)三、解答題12已知f(x)log2(x1)，當(dāng)點(diǎn)(x，y)在函數(shù)yf(x)的圖象上時(shí)，點(diǎn)在函數(shù)yg(x)的圖象上(1)寫出yg(x)的解析式；(2)求方程f(x)g(x)0的根考點(diǎn)對(duì)數(shù)函數(shù)的解析式題點(diǎn)對(duì)數(shù)函數(shù)的解析式解(1)設(shè)x，y，則x3x，y2y.(x，y)在yf(x)的圖象上，ylog2(x1)，2ylog2(3x1)，ylog2(3x1)，即點(diǎn)(x，y)在ylog2(3x1)的圖象上g(x)log2(3x1)(2)f(x)g(x)0，即log2(x1)log2(3x1)log2，x1，解得x0或x1.13已知1x4，求函數(shù)f(x)log2log2的最大值與最小值考點(diǎn)對(duì)數(shù)函數(shù)的值域題點(diǎn)對(duì)數(shù)函數(shù)的值域解f(x)log2log2(log2x2)(log2x1)2，又1x4，0log2x2，當(dāng)log2x，即x22時(shí)，f(x)取最小值；當(dāng)lo