激趣誘思蘇格蘭數(shù)學(xué)家納皮爾在研究天文學(xué)的過(guò)程中,為了簡(jiǎn)化其中的運(yùn)算而發(fā)明了對(duì)數(shù).對(duì)數(shù)的發(fā)明是數(shù)學(xué)史上的重大事件,恩格斯曾經(jīng)把對(duì)數(shù)的發(fā)明、解析幾何的創(chuàng)始、微積分的建立并稱(chēng)為17世紀(jì)數(shù)學(xué)的三大成就.伽利略也說(shuō)過(guò):“給我空間、時(shí)間及對(duì)數(shù),我就可以創(chuàng)造一個(gè)宇宙.”對(duì)數(shù)究竟是什么?它為何有如此大的魅力?它的作用何在?知識(shí)點(diǎn)撥一、對(duì)數(shù)的概念1.一般地,如果a(a>0,且a≠1)的b次冪等于N,即ab=N,那么數(shù)b稱(chēng)為以a為底N的對(duì)數(shù),記作logaN=b,其中a叫作對(duì)數(shù)的底數(shù),N叫作真數(shù).要點(diǎn)筆記

“l(fā)og”同+、-、×等符號(hào)一樣,表示一種運(yùn)算,即已知一個(gè)底數(shù)和它的冪求指數(shù)的運(yùn)算,這種運(yùn)算叫對(duì)數(shù)運(yùn)算,不過(guò)對(duì)數(shù)運(yùn)算的符號(hào)寫(xiě)在數(shù)的前面.2.兩種特殊的對(duì)數(shù):名稱(chēng)定義常用對(duì)數(shù)當(dāng)對(duì)數(shù)的底數(shù)a=10時(shí),通常稱(chēng)之為常用對(duì)數(shù),N的常用對(duì)數(shù)log10N,簡(jiǎn)記為lgN自然對(duì)數(shù)在科學(xué)領(lǐng)域,常常使用無(wú)理數(shù)e=2.718281…為底數(shù)的對(duì)數(shù),稱(chēng)之為自然對(duì)數(shù),并將logeN簡(jiǎn)記為lnN.微拓展給定底數(shù)后,對(duì)數(shù)運(yùn)算是指數(shù)運(yùn)算的逆運(yùn)算.微練習(xí)

(2)已知log4x=2,則x等于(

)A.±4 B.4C.16 D.2答案(1)B

(2)C二、對(duì)數(shù)的基本性質(zhì)1.負(fù)數(shù)和零沒(méi)有對(duì)數(shù).2.對(duì)于任意的a>0,且a≠1,都有l(wèi)oga1=0,logaa=1,名師點(diǎn)析

1.loga1=0,logaa=1可簡(jiǎn)述為“1的對(duì)數(shù)等于0,底的對(duì)數(shù)等于1”.2.對(duì)數(shù)恒等式的特點(diǎn):(1)指數(shù)中含有對(duì)數(shù)形式;(2)同底,即冪底數(shù)和對(duì)數(shù)的底數(shù)相同;(3)其值為對(duì)數(shù)的真數(shù).(2)若log3(log2x)=0,則x=

.

解析(2)由已知得log2x=1,故x=2.答案(1)D

(2)2課堂篇探究學(xué)習(xí)探究一對(duì)數(shù)式與指數(shù)式的互化例1將下列指數(shù)式與對(duì)數(shù)式互化:分析利用當(dāng)a>0,且a≠1時(shí),logaN=b?ab=N進(jìn)行互化.反思感悟

1.logaN=b(a>0,且a≠1)與ab=N(a>0,且a≠1)是等價(jià)的,表示a,b,N三者之間的同一種關(guān)系.如下表:式子名稱(chēng)意義abN指數(shù)式ab=N底數(shù)指數(shù)冪a的b次冪等于N對(duì)數(shù)式logaN=b底數(shù)對(duì)數(shù)真數(shù)以a為底N的對(duì)數(shù)等于b2.將指數(shù)式化為對(duì)數(shù)式,只需將冪作為真數(shù),指數(shù)作為對(duì)數(shù),底數(shù)不變;而將對(duì)數(shù)式化為指數(shù)式,只需將對(duì)數(shù)式的真數(shù)作為冪,對(duì)數(shù)作為指數(shù),底數(shù)不變.變式訓(xùn)練

1將下列指數(shù)式與對(duì)數(shù)式互化:探究二利用對(duì)數(shù)式與指數(shù)式的關(guān)系求值例2求下列各式中x的值:(1)4x=5·3x;

(2)log7(x+2)=2;(3)lne2=x;

(4)logx27=;(5)lg0.01=x.分析利用指數(shù)式與對(duì)數(shù)式之間的關(guān)系求解.要點(diǎn)筆記

指數(shù)式ax=N(a>0,且a≠1)與對(duì)數(shù)式x=logaN(a>0,且a≠1)表示了三個(gè)量a,x,N之間的同一種關(guān)系,因而已知其中兩個(gè)時(shí),可以通過(guò)對(duì)數(shù)式與指數(shù)式的相互轉(zhuǎn)化求出第三個(gè).變式訓(xùn)練

2求下列各式中的x值:(1)log2x=;(2)log216=x;(3)logx27=3.(2)∵log216=x,∴2x=16.∴2x=24.∴x=4.(3)∵logx27=3,∴x3=27.即x3=33.∴x=3.探究三利用對(duì)數(shù)的基本性質(zhì)與對(duì)數(shù)恒等式求值例3求下列各式中x的值:(1)ln(log2x)=0;

(2)log2(lgx)=1;分析利用logaa=1,loga1=0(a>0,且a≠1)及對(duì)數(shù)恒等式求值.解(1)∵ln(log2x)=0,∴l(xiāng)og2x=1.∴x=21=2.(2)∵log2(lg

x)=1,∴l(xiāng)g

x=2.∴x=102=100.反思感悟

1.在對(duì)數(shù)的運(yùn)算中,常見(jiàn)的對(duì)數(shù)的基本性質(zhì)有:(1)負(fù)數(shù)和零沒(méi)有對(duì)數(shù);(2)loga1=0(a>0,且a≠1);(3)logaa=1(a>0,且a≠1).2.對(duì)指數(shù)中含有對(duì)數(shù)的式子進(jìn)行化簡(jiǎn)、求值時(shí),應(yīng)充分考慮對(duì)數(shù)恒等式的應(yīng)用.對(duì)數(shù)恒等式

=N(a>0,且a≠1,N>0)的結(jié)構(gòu)特點(diǎn)是:(1)指數(shù)中含有對(duì)數(shù);(2)它們是同底的;(3)其值為對(duì)數(shù)的真數(shù).變式訓(xùn)練

3求下列各式中x的值:(1)ln(lgx)=1;(2)log2(log5x)=0;解(1)∵ln(lg

x)=1,∴l(xiāng)g

x=e.∴x=10e.(2)∵log2(log5x)=0,∴l(xiāng)og5x=1.∴x=5.當(dāng)堂檢測(cè)1.將log5b=2化為指數(shù)式是(

)A.5b=2 B.b5=2 C.52=b D.b2=5答案C2.(2020全國(guó)1,文8)設(shè)alog34=2,