2023版高考數(shù)學(xué)一輪總復(fù)習(xí)第二章函數(shù)概念與基本初等函數(shù)Ⅰ第五講對(duì)數(shù)與對(duì)數(shù)函數(shù)課件理

第二章函數(shù)概念與基本初等函數(shù)Ⅰ第五講對(duì)數(shù)與對(duì)數(shù)函數(shù)要點(diǎn)提煉

對(duì)數(shù)與對(duì)數(shù)運(yùn)算考點(diǎn)11.對(duì)數(shù)的概念一般地,如果ax=N(a>0,且a≠1),那么數(shù)x叫作以a為底N的對(duì)數(shù),記作

,其中a叫作對(duì)數(shù)的底數(shù),N叫作真數(shù).由此可得對(duì)數(shù)式與指數(shù)式的互化:ax=N?logaN=x(a>0,且a≠1).說明

幾種常見的對(duì)數(shù)對(duì)數(shù)形式特點(diǎn)記法一般對(duì)數(shù)底數(shù)為a(a>0,且a≠1)logaN常用對(duì)數(shù)底數(shù)為10lgN自然對(duì)數(shù)底數(shù)為elnNx=logaN

對(duì)數(shù)與對(duì)數(shù)運(yùn)算考點(diǎn)12.對(duì)數(shù)的性質(zhì)、運(yùn)算法則及換底公式性質(zhì)

運(yùn)算法則換底公式說明

(1)應(yīng)用換底公式時(shí),一般選用e或10作為底數(shù).(2)表中有關(guān)公式均是在式子中所有對(duì)數(shù)有意義的前提下成立的.NxlogaM+logaNlogaM-logaNnlogaM

對(duì)數(shù)函數(shù)的圖象與性質(zhì)考點(diǎn)21.對(duì)數(shù)函數(shù)的圖象和性質(zhì)

a>10<a<1圖象性質(zhì)定義域:

.值域:R.圖象過定點(diǎn)

,即恒有l(wèi)oga1=0.當(dāng)x>1時(shí),恒有y>0;當(dāng)0<x<1時(shí),恒有y<0.當(dāng)x>1時(shí),恒有y<0;當(dāng)0<x<1時(shí),恒有y>0.在(0,+∞)上是增函數(shù).

在(0,+∞)上是

.

(0,+∞)(1,0)減函數(shù)對(duì)數(shù)函數(shù)的圖象與性質(zhì)考點(diǎn)22.對(duì)數(shù)函數(shù)圖象的特點(diǎn)

注意

當(dāng)對(duì)數(shù)函數(shù)的底數(shù)a的大小不確定時(shí),需分a>1和0<a<1兩種情況進(jìn)行討論.增大

對(duì)數(shù)函數(shù)的圖象與性質(zhì)考點(diǎn)23.反函數(shù)指數(shù)函數(shù)y=ax(a>0,且a≠1)與對(duì)數(shù)函數(shù)y=logax(a>0,且a≠1)互為反函數(shù),它們的圖象關(guān)于直線

對(duì)稱(如圖所示).反函數(shù)的定義域、值域分別是原函數(shù)的值域、定義域,互為反函數(shù)的兩個(gè)函數(shù)具有相同的單調(diào)性、奇偶性.

y=x

??√√2

考向掃描

對(duì)數(shù)式的運(yùn)算考向11.典例(1)[2020全國卷Ⅰ]設(shè)alog34=2,則4-a=(

)A.116 B.19 C.18 D.16(2)[2018全國卷Ⅲ][理]設(shè)a=log0.20.3,b=log20.3,則(

)A.a+b<ab<0 B.ab<a+b<0C.a+b<0<ab

D.ab<0<a+b(3)[2018全國卷Ⅰ]已知函數(shù)f(x)=log2(x2+a).若f(3)=1,則a=

.

BB-7

對(duì)數(shù)式的運(yùn)算考向1

對(duì)數(shù)式的運(yùn)算考向1

42

對(duì)數(shù)函數(shù)的圖象及應(yīng)用考向2

A B C DD

對(duì)數(shù)函數(shù)的圖象及應(yīng)用考向2

對(duì)數(shù)函數(shù)的圖象及應(yīng)用考向24.變式[2021長春市第四次質(zhì)量監(jiān)測(cè)]如圖,①②③④中不屬于函數(shù)y=log2x,y=log0.5x,y=-log3x對(duì)應(yīng)圖象的是(

)A.① B.② C.③ D.④

解析

因?yàn)閥=log0.5x=-log2x,所以函數(shù)y=log2x與y=log0.5x的圖象關(guān)于x軸對(duì)稱,又當(dāng)x>1時(shí),log2x>log3x,所以當(dāng)x>1時(shí),-log2x<-log3x,所以函數(shù)y=log2x,y=log0.5x,y=-log3x對(duì)應(yīng)的圖象分別為①④③,故選B.B

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3角度1比較大小5.典例[2020全國卷Ⅲ][理]已知55<84,134<85.設(shè)a=log53,b=log85,c=log138,則(

)A.a<b<c

B.b<a<cC.b<c<a

D.c<a<bA

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3方法技巧比較對(duì)數(shù)值大小的常見類型及解題方法常見類型解題方法底數(shù)為同一常數(shù)可由對(duì)數(shù)函數(shù)的單調(diào)性直接進(jìn)行判斷底數(shù)為同一字母需對(duì)底數(shù)進(jìn)行分類討論底數(shù)不同,真數(shù)相同可以先用換底公式化為同底后,再進(jìn)行比較底數(shù)與真數(shù)都不同常借助1,0等中間量進(jìn)行比較

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3角度2解對(duì)數(shù)方程或不等式

C

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3

注意

化為同底數(shù)的對(duì)數(shù)時(shí),可以借助對(duì)數(shù)的性質(zhì)、運(yùn)算法則及換底公式進(jìn)行.

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3角度3對(duì)數(shù)型函數(shù)的單調(diào)性問題

D

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3方法技巧對(duì)數(shù)型復(fù)合函數(shù)的單調(diào)性問題的求解策略1.對(duì)于y=loga

f(x)型的復(fù)合函數(shù)的單調(diào)性,有以下結(jié)論:函數(shù)y=loga

f(x)的單調(diào)性與函數(shù)u=f(x)(f(x)>0)的單調(diào)性在a>1時(shí)相同,在0<a<1時(shí)相反.2.研究y=f(logax)型的復(fù)合函數(shù)的單調(diào)性,一般用換元法,即令t=logax,則只需研究t=logax及y=f(t)的單調(diào)性即可.注意

研究對(duì)數(shù)型復(fù)合函數(shù)的單調(diào)性,一定要堅(jiān)持“定義域優(yōu)先”原則,否則所得范圍易出錯(cuò).

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向38.變式(1)[2019天津高考][理]已知a=log52,b=log0.50.2,c=0.50.2,則a,b,c的大小關(guān)系為(

)

A.a<c<b

B.a<b<cC.b<c<a

D.c<a<b(2)[2017全國卷Ⅰ]已知函數(shù)f(x)=lnx+ln(2-x),則(

)A.f(x)在(0,2)上單調(diào)遞增B.f(x)在(0,2)上單調(diào)遞減C.y=f(x)的圖象關(guān)于直線x=1對(duì)稱D.y=f(x)的圖象關(guān)于點(diǎn)(1,0)對(duì)稱AC

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3

對(duì)數(shù)函數(shù)的性質(zhì)及應(yīng)用考向3

指數(shù)函數(shù)、對(duì)數(shù)函數(shù)的綜合問題考向49.典例[2020全國卷Ⅰ][理]若2a+log2a=4b+2log4b,則(

)A.a>2b

B.a<2b

C.a>b2

D.a<b2

解析

令f(x)=2x+log2x,因?yàn)閥=2x在(0,+∞)上單調(diào)遞增,y=log2x在(0,+∞)上單調(diào)遞增,所以f(x)=2x+log2x在(0,+∞)上單調(diào)遞增.又2a+log2a=4b+2log4b=22b+log2b<22b+log2(2b),(放縮)所以f(a)<f(2b),所以a<2b.故選B.B

指數(shù)函數(shù)、對(duì)數(shù)函數(shù)的綜合問題考向410.變式[2019全國卷Ⅱ][理]已知f(x)是奇函數(shù),且當(dāng)x<0時(shí),f(x)=-eax.若f(ln2)=8,則a=

.

-3攻堅(jiān)克難指數(shù)、對(duì)數(shù)比較大小的策略數(shù)學(xué)探索策略1利用指數(shù)、對(duì)數(shù)函數(shù)的圖象與性質(zhì)比較數(shù)(式)的大小11.典例[2021蘇錫常鎮(zhèn)四市聯(lián)考]已知a=sin1,b=cos1,則下列不等式正確的是（）

A.logab<ab<ba B.logab<ba<abC.ab<ba<logab D.ba<ab<logabD指數(shù)、對(duì)數(shù)比較大小的策略數(shù)學(xué)探索

指數(shù)、對(duì)數(shù)比較大小的策略數(shù)學(xué)探索策略2

涉及三元變量的比較大小問題12.典例[2017全國卷Ⅰ][理]設(shè)x,y,z為正數(shù),且2x=3y=5z,則（）A.2x<3y<5z B.5z<2x<3yC.3y<5z