高中數(shù)學(xué)第二章2.2對(duì)數(shù)函數(shù)2.2.1對(duì)數(shù)與對(duì)數(shù)運(yùn)算（第2課時(shí)）學(xué)案（含解析）新人教A版.docx

第2課時(shí)對(duì)數(shù)的運(yùn)算學(xué)習(xí)目標(biāo)1.掌握積、商、冪的對(duì)數(shù)運(yùn)算性質(zhì)，理解其推導(dǎo)過(guò)程和成立條件.2.掌握換底公式及其推論.3.能熟練運(yùn)用對(duì)數(shù)的運(yùn)算性質(zhì)進(jìn)行化簡(jiǎn)求值知識(shí)點(diǎn)一對(duì)數(shù)運(yùn)算性質(zhì)如果a0，且a1，M0，N0，那么：(1)loga(MN)logaMlogaN；(2)logalogaMlogaN；(3)logaMnnlogaM(nR)知識(shí)點(diǎn)二換底公式思考1觀察知識(shí)點(diǎn)一的三個(gè)公式，我們發(fā)現(xiàn)對(duì)數(shù)都是同底的才能用這三個(gè)公式那么我們?cè)谶\(yùn)算和求值中遇到不同底的對(duì)數(shù)怎么辦？答案設(shè)法換為同底梳理對(duì)數(shù)換底公式：logab(a0，且a1，b0，c0，且c1)特別地：logablogba1(a0，且a1，b0，且b1)1log2x22log2x.()2loga(2)(3)loga(2)loga(3)()3logaMlogaNloga(MN)()4logx2.()類型一具體數(shù)字的化簡(jiǎn)求值例1計(jì)算：(1)log345log35；(2)log2(2345)；(3)；(4)log29log38.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)解(1)log345log35log3log39log3322log332.(2)log2(2345)log2(23210)log2(213)13log2213.(3)原式=.(4)log29log38log2(32)log3(23)2log233log326log236.反思與感悟具體數(shù)的化簡(jiǎn)求值主要遵循兩個(gè)原則(1)把數(shù)字化為質(zhì)因數(shù)的冪、積、商的形式(2)不同底化為同底跟蹤訓(xùn)練1計(jì)算：(1)2log63log64；(2)(lg25lg)(3)log43log98；(4)log2.56.25ln考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)指數(shù)對(duì)數(shù)的混合運(yùn)算解(1)原式log632log64log6(324)log6(62)2log662.(2)原式lg10210121020.(3)原式.(4)原式log2.5(2.5)22.類型二代數(shù)式的化簡(jiǎn)命題角度1代數(shù)式恒等變換例2化簡(jiǎn)loga.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)解0且x20，0，y0，z0.logaloga(x2)logalogax2logaloga2loga|x|logaylogaz.反思與感悟使用公式要注意成立條件，如lgx2不一定等于2lgx，反例：log10(10)22log10(10)是不成立的要特別注意loga(MN)logaMlogaN，loga(MN)logaMlogaN.跟蹤訓(xùn)練2已知y0，化簡(jiǎn)loga.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)解0，y0，x0，z0.logalogaloga(yz)logaxlogaylogaz.命題角度2用代數(shù)式表示對(duì)數(shù)例3已知log189a,18b5，求log3645.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)用代數(shù)式表示對(duì)數(shù)解方法一log189a,18b5，log185b，于是log3645.方法二log189a,18b5，log185b，于是log3645.方法三log189a,18b5，lg9alg18，lg5blg18，log3645.反思與感悟此類問(wèn)題的本質(zhì)是把目標(biāo)分解為基本“粒子”，然后用指定字母換元跟蹤訓(xùn)練3已知log23a，log37b，用a，b表示log4256.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)用代數(shù)式表示對(duì)數(shù)解log23a，則log32，又log37b，log4256.1log5log53等于()A0B1C1Dlog5考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)答案A2設(shè)a，b，c均為不等于1的正實(shí)數(shù)，則下列等式中恒成立的是()AlogablogcblogcaBlogablogcalogcbCloga(bc)logablogacDloga(bc)logablogac考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)換底公式的應(yīng)用答案B解析由logablogcblogca，故A錯(cuò)；由logablogcalogcb.故選B.3log29log34等于()A.B.C2D4考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)換底公式的應(yīng)用答案D4lg0.01log216的值是_考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)答案2解析lg0.01log216242.5若2x3y，則_.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)用代數(shù)式表示對(duì)數(shù)答案log23解析方法一設(shè)2x3yt，則xlog2t，ylog3t.log23.方法二2x3y，則lg2xlg3y，xlg2ylg3，log23.1換底公式可完成不同底數(shù)的對(duì)數(shù)式之間的轉(zhuǎn)化，可正用、逆用；使用的關(guān)鍵是恰當(dāng)選擇底數(shù)，換底的目的是利用對(duì)數(shù)的運(yùn)算性質(zhì)進(jìn)行對(duì)數(shù)式的化簡(jiǎn)2運(yùn)用對(duì)數(shù)的運(yùn)算性質(zhì)應(yīng)注意：(1)在各對(duì)數(shù)有意義的前提下才能應(yīng)用運(yùn)算性質(zhì)(2)根據(jù)不同的問(wèn)題選擇公式的正用或逆用(3)在運(yùn)算過(guò)程中避免出現(xiàn)以下錯(cuò)誤：logaNn(logaN)n，loga(MN)logaMlogaN，logaMlogaNloga(MN)一、選擇題1下列各式(各式均有意義)不正確的個(gè)數(shù)為()loga(MN)logaMlogaN；loga(MN)；a；(am)namn；nlogab.A2B3C4D5考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)答案B解析正確，不正確，正確，不正確，不正確2如果lgxlga3lgb5lgc，那么()AxBxCxa3b5cDxab3c3考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)答案A解析lga3lgb5lgclgalgb3lgc5lg，由lgxlg，可得x.3log4等于()A.B.C2D4考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)答案D解析log4log()44.4化簡(jiǎn)等于()Alog54B3log52C2D3考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)換底公式的應(yīng)用答案D解析log28log2233.5已知lg2a，lg3b，則用a，b表示lg15為()Aba1Bb(a1)Cba1Db(1a)考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)用代數(shù)式表示對(duì)數(shù)答案A解析lg15lg(35)lg3lg5lg3lglg31lg2ba1.6若log5log36log6x2，則x等于()A9B.C25D.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)換底公式的應(yīng)用答案D解析由換底公式，得2，lgx2lg5，x52.7計(jì)算(log32log23)2的值是()Alog26Blog36C2D1考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)答案C解析原式(log32)22log32log23(log23)2(log32)2(log23)22.8化簡(jiǎn)：log2等于()A2B22log23C2D2log232考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)答案B解析|log232|2log23.原式2log23log22log23log2322log23.二、填空題9(log43log83)(log32log92)_.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)換底公式的應(yīng)用答案解析原式log23.10(lg5)2lg2lg50_.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)利用lg2lg51化簡(jiǎn)求解對(duì)數(shù)值答案1解析(lg5)2lg2lg50(lg5)2lg2(lg5lg10)(lg5)2lg2lg5lg2lg5(lg5lg2)lg2lg5lg21.11若3x4y36，則_.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)用代數(shù)式表示對(duì)數(shù)答案1解析3x4y36，兩邊取以6為底的對(duì)數(shù)，得xlog63ylog642，log63，log64，即log62，故log63log621.三、解答題12計(jì)算：(1)(log33)2log0.259log5log1；(2).考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)對(duì)數(shù)的運(yùn)算性質(zhì)解(1)(log33)2log0.259log5log121901.(2)1.13已知x，y，z為正數(shù)，3x4y6z,2xpy.(1)求p的值；(2)求證：.考點(diǎn)對(duì)數(shù)的運(yùn)算題點(diǎn)用代數(shù)式表示對(duì)數(shù)(1)解設(shè)3x4y6zt，則t0，且t1.xlog3t，ylog4t，zlog6t.2xpy，2log3tplog4tp.log3t0，p2log344log32.(2)證明logt6logt3logt2.又lo