高職高考數(shù)學(xué)復(fù)習(xí)§4.3對(duì)數(shù)的概念及運(yùn)算【復(fù)習(xí)目標(biāo)】1.掌握對(duì)數(shù)概念,能熟練地進(jìn)行指數(shù)式和對(duì)數(shù)式的互化.2.掌握積、商、冪的對(duì)數(shù)和對(duì)數(shù)的換底公式,能用公式進(jìn)行化簡(jiǎn)、求值、證明.【知識(shí)回顧】1.對(duì)數(shù)概念如果ab=N(a>0且a≠1)，那么b叫做以a為底N的對(duì)數(shù)，記作logaN=b，其中a叫做底，N叫做真數(shù).特別地，以10為底的對(duì)數(shù)叫做常用對(duì)數(shù)，log10N可簡(jiǎn)記作lg

N.以無(wú)理數(shù)e=2.71828…為底的對(duì)數(shù)叫做自然對(duì)數(shù)，logeN可簡(jiǎn)記作lnN.2.對(duì)數(shù)的性質(zhì)(1)1的對(duì)數(shù)等于零,即loga1=0(a>0且a≠1).(2)底的對(duì)數(shù)等于1,即logaa=1(a>0且a≠1).(3)零和負(fù)數(shù)沒有對(duì)數(shù).

【點(diǎn)評(píng)】根據(jù)指數(shù)式與對(duì)數(shù)式互化方法y=ax?x=logay(x∈R,y>0,a>0且a≠1)解答.

【點(diǎn)評(píng)】利用積、商、冪的對(duì)數(shù)運(yùn)算公式解答.【對(duì)點(diǎn)練習(xí)2】計(jì)算:(1)lg(100×103)=

;

(2)log36-log34+log318=

.

【答案】(1)5

(2)3【例3】已知lg2=p,lg3=q,用p,q表示下列各對(duì)數(shù). (1)lg8; (2)lg36.【解】(1)lg8=lg23=3lg2=3p.(2)lg36=lg4+lg9=2lg2+2lg3=2p+2q.【點(diǎn)評(píng)】利用對(duì)數(shù)運(yùn)算法則解答.【對(duì)點(diǎn)練習(xí)3】已知lg2=a,lg3=b,用a,b表示下列各對(duì)數(shù).(1)lg9=

; (2)lg6=

.

【答案】(1)2b

(2)a+b【例4】解下列方程: (1)log3x=0; (2)log2(x-1)=1.【解】(1)由log3x=0,得x=30=1.經(jīng)檢驗(yàn),x=1是原方程的解.(2)由log2(x-1)=1,得x-1=21,則x=3.經(jīng)檢驗(yàn),x=3是原方程的解.【點(diǎn)評(píng)】根據(jù)對(duì)數(shù)的定義及性質(zhì)解答.【對(duì)點(diǎn)練習(xí)4】已知log2(x2-3)=0,則x=

.

【答案】2【仿真訓(xùn)練】一、選擇題1.將32=9改寫成對(duì)數(shù)形式是 (

) A.log23=9 B.log32=9 C.log39=2 D.log93=2【答案】C

【答案】C

【答案】D

【答案】C5.計(jì)算:log62+log63= (

) A.6 B.5 C.1 D.log65【答案】C

8.設(shè)x>0,y>0,a>0且a≠1,則下列等式中,正確的是 (

) A.(ax)y=axy

B.loga(x+y)=logax+logay

C.axy=axay

D.loga(xy)=logax·logay

13.計(jì)算:log220-log25=

.

【答案】214.已知log3x=2,則x=

.

【答案】915.若lg2=a,則lg4=

;lg5=

.

【答案】2a

1-a

三、解答題16.計(jì)算: (1)log2(47×25); (2)log315+log318-log310.【解】(1)log2(47×25)=log2(214×25)=log2219=19.

(2)log315+log318-log310=log3(15×18÷10)=log327=3.17.解下列方程: (1)log5x=0; (2)log2(x+3)=1.【解】(1)由log5x=