對(duì)數(shù)與對(duì)數(shù)函數(shù)講義

1、 對(duì)數(shù)與對(duì)數(shù)函數(shù)（1） 對(duì)數(shù)1對(duì)數(shù)的概念：一般地，如果，那么數(shù)叫做以為底的對(duì)數(shù)，記作：（ 底數(shù)， 真數(shù)， 對(duì)數(shù)式）說明： 注意底數(shù)的限制，且； ； 注意對(duì)數(shù)的書寫格式兩個(gè)重要對(duì)數(shù)： 常用對(duì)數(shù)：以10為底的對(duì)數(shù)； 自然對(duì)數(shù)：以無理數(shù)為底的對(duì)數(shù)的對(duì)數(shù)指數(shù)式與對(duì)數(shù)式的互化 冪值 真數(shù) n b 底數(shù) 指數(shù) 對(duì)數(shù)（二）對(duì)數(shù)的運(yùn)算性質(zhì)如果，且，那么： ； ； 注意：換底公式（，且；，且；）利用換底公式推導(dǎo)下面的結(jié)論（1）；（2）（二）對(duì)數(shù)函數(shù)1、對(duì)數(shù)函數(shù)的概念：函數(shù)，且叫做對(duì)數(shù)函數(shù)，其中是自變量，函數(shù)的定義域是（0，+）注意： 對(duì)數(shù)函數(shù)的定義與指數(shù)函數(shù)類似，都是形式定義，注意辨別。如：， 都不是對(duì)數(shù)函數(shù)，

2、而只能稱其為對(duì)數(shù)型函數(shù) 對(duì)數(shù)函數(shù)對(duì)底數(shù)的限制：，且2、對(duì)數(shù)函數(shù)的性質(zhì)：a10a1定義域x0定義域x0值域?yàn)閞值域?yàn)閞在r上遞增在r上遞減函數(shù)圖象都過定點(diǎn)（1，0）函數(shù)圖象都過定點(diǎn)（1，0）題型一：對(duì)數(shù)的定義與對(duì)數(shù)運(yùn)算【例1】 將下列指數(shù)式化為對(duì)數(shù)式，對(duì)數(shù)式化為指數(shù)式：；.求下列各式中的值：；.【例2】 將下列指數(shù)式化為對(duì)數(shù)式，對(duì)數(shù)式化為指數(shù)式：（1）； （2）； （3）；（4）； （5）； （6）ln100=4.606.【例3】 將下列對(duì)數(shù)式寫成指數(shù)式：（1）；（2）；（3）； （4）【例4】 已知, 則的值等于（ ）.a. 1 b. 2 c. 8 d. 12【例5】 計(jì)算下列各式的值：（1）

3、； （2）； （3）.【例6】 ，【例7】 （）等于（ ）.a. 1 b. 1c. 2d. 2【例8】 （a0）化簡(jiǎn)得結(jié)果是（）.a. ab. a2c. a d. a【例9】 化簡(jiǎn)的結(jié)果是（ ）. a. b. 1 c. 2 d.【例10】 計(jì)算 . 【例11】 計(jì)算：=_【例12】 化簡(jiǎn)與求值：；【例13】 若，則= . 【例14】 化簡(jiǎn)的結(jié)果是 （ ）. a .1 b. c. 2 d.3 【例15】 計(jì)算： 5 【例16】 求下列各值：；.【例17】 設(shè)，求實(shí)數(shù)m的值【例18】 （1）設(shè)，求的值.（2）設(shè),，且，求a的值.題型二：對(duì)數(shù)運(yùn)算法則的應(yīng)用【例19】 若、，且、，則a.b.c.或d.

4、、為一切非1的正數(shù)【例20】 下列各式中，正確的是（ ）a.b.c.d.【例21】 已知求證：【例22】 已知，用 a 表示【例23】 若，則 . 【例24】 已知，用表示【例25】 已知且，則等于a.b.c.d.【例26】 已知，用表示.【例27】 （1）已知，試用a、b表示的值；（2）已知，用a、b表示. 【例28】 已知，求題型三：對(duì)數(shù)方程【例29】 求底數(shù)：（1）, （2）【例30】 已知，求實(shí)數(shù)的值.【例31】 證明：【例32】 求x的值： 【例33】 解方程【例34】 方程的解x= ；【例35】 設(shè)a為實(shí)常數(shù)，解關(guān)于x的方程.【例36】 設(shè)正數(shù)a，b，c滿足.（1）求證：；（2）又

5、設(shè)，求a，b，c的值.題型一 對(duì)數(shù)函數(shù)的基本性質(zhì)【例1】 下面結(jié)論中，不正確的是 a.若a1，則與在定義域內(nèi)均為增函數(shù)b.函數(shù)與圖象關(guān)于直線對(duì)稱c.與表示同一函數(shù)d.若，則一定有【例2】 圖中的曲線是的圖象，已知的值為，則相應(yīng)曲線的依次為（ ）. a. ， b. ， c. ， d. ，0xc1c2c4c31y【例3】 當(dāng)時(shí),在同一坐標(biāo)系中,函數(shù)的圖象是（ ）. xy11oxyo11oyx11oyx11 a b c d【例4】 設(shè)，函數(shù)在區(qū)間上的最大值與最小值之差為，則（ ）. a.b. 2c. d. 4【例5】 若，則a的取值范圍是 a.b.c.d.或a1【例6】 比較兩個(gè)對(duì)數(shù)值的大?。?；

6、.【例7】 若，那么滿足的條件是（ ）. a. b. c. d. 【例8】 已知，則（）a.b.c.d.【例9】 下列各式錯(cuò)誤的是（ ）. a. b. c. d. .【例10】 若，則（）a.b.c.d.【例11】 若，求的關(guān)系?！纠?2】 已知，求的取值范圍.題型二 對(duì)數(shù)型符合型復(fù)合函數(shù)的定義域值域【例13】 下列函數(shù)中哪個(gè)與函數(shù)y=x是同一個(gè)函數(shù)（ ） a. b. y= c. d. y=【例14】 函數(shù)的定義域是（ ）. a. b. c. d. 【例15】 函數(shù)的定義域?yàn)?. （用區(qū)間表示）【例16】 求下列函數(shù)的定義域：（1） (2)【例17】 求下列函數(shù)的定義域：；.【例18】 求下列

7、函數(shù)的定義域：；.【例19】 求下列函數(shù)的定義域： 【例20】 求下列函數(shù)的定義域：（1） ； （2）.【例21】 函數(shù)的值域是（ ）. a. r b. c. d. 【例22】 函數(shù)的值域是 待添加的隱藏文字內(nèi)容3a.y0b.yrc.y0且y1d.y2【例23】 求下列函數(shù)的定義域、值域：1234題型三 對(duì)數(shù)型符合型復(fù)合函數(shù)的單調(diào)性【例24】 下列函數(shù)中，在上為增函數(shù)的是（ ）. a. b. c. d. 【例25】 證明函數(shù)y= (+1)在（0，+）上是減函數(shù)；【例26】 討論函數(shù)的單調(diào)性.【例27】 求的單調(diào)遞減區(qū)間【例28】 求函數(shù)的單調(diào)遞增區(qū)間【例29】 求函數(shù)的單調(diào)區(qū)間【例30】 求函

8、數(shù)的單調(diào)區(qū)間題型四 對(duì)數(shù)函數(shù)的綜合與應(yīng)用【例31】 函數(shù)是 函數(shù). （填“奇”、“偶”或“非奇非偶”）【例32】 如果，求的取值范圍.【例33】 已知，要使ab，求實(shí)數(shù)k的取值范圍.【例34】 已知函數(shù)，求：（1）的值域； （2）的最大值及相應(yīng)x的值.【例35】 已知函數(shù)其中（1）求函數(shù)的定義域； （2）判斷的奇偶性，并說明理由；winger tuivasa-sheck, who scored two tries in the kiwis 20-18 semi-final win over england, has been passed fit after a lower-leg injur

9、y, while slater has been named at full-back but is still recovering from a knee injury aggravated against usa.both sides boast 100% records heading into the encounter but australia have not conceded a try since josh charnleys effort in their first pool match against england on the opening day.aussie

10、 winger jarryd hayne is the competitions top try scorer with nine, closely followed by tuivasa-sheck with eight.but it is recently named rugby league international federation player of the year sonny bill williams who has attracted the most interest in the tournament so far.the kiwi - with a tournam

11、ent high 17 offloads - has the chance of becoming the first player to win the world cup in both rugby league and rugby union after triumphing with the all blacks in 2011.id give every award back in a heartbeat just to get across the line this weekend, said williams.the (lack of) air up there watch m

12、cayman islands-based webb, the head of fifas anti-racism taskforce, is in london for the football associations 150th anniversary celebrations and will attend citys premier league match at chelsea on sunday.i am going to be at the match tomorrow and i have asked to meet yaya toure, he told bbc sport.

13、for me its about how he felt and i would like to speak to him first to find out what his experience was.uefa hasopened disciplinary proceedings against cskafor the racist behaviour of their fans duringcitys 2-1 win.michel platini, president of european footballs governing body, has also ordered an i

14、mmediate investigation into the referees actions.cska said they were surprised and disappointed by toures complaint. in a statement the russian side added: we found no racist insults from fans of cska. baumgartner the disappointing news: mission aborted.the supersonic descent could happen as early a

15、s sunda.the weather plays an important role in this mission. starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. the balloon, with capsule attached, will move through the lower level of the atmosphere (the tropos

16、phere) where our day-to-day weather lives. it will climb higher than the tip of mount everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. as he crosses the boundary layer (called the tropo

17、pause),e can expect a lot of turbulence.the balloon will slowly drift to the edge of space at 120,000 feet ( then, i would assume, he will slowly step out onto something resembling an olympic diving platform.they blew it in 2008 when they got caught cold in the final and they will not make the same

18、mistake against the kiwis in manchester.five years ago they cruised through to the final and so far history has repeated itself here - the last try they conceded was scored by englands josh charnley in the opening game of the tournament.that could be classed as a weakness, a team under-cooked - but

19、i have been impressed by the kangaroos focus in their games since then.they have been concentrating on the sort of stuff that wins you tough, even contests - strong defence, especially on their own goal-line, completing sets and a good kick-chase. theyve been great at all the unglamorous stuff that

20、often goes unnoticed in the stands but not by your team-mates.it is as though their entire tournament has been preparation for the final.in johnathan thurston, cooper cronk, cameron smith and either billy slater or greg inglis at full-back they have a spine that is unmatched in rugby league. they ha

21、ve played in so many high-pressure games - a priceless asset going into saturday.the kiwis are a lot less experienced but winning a dramatic match like their semi-final against england will do wonders for their confidence.they defeated australia in the four nations final in 2010 and the last world c

22、up, and know they can rise to the big occasion.winger tuivasa-sheck, who scored two tries in the kiwis 20-18 semi-final win over england, has been passed fit after a lower-leg injury, while slater has been named at full-back but is still recovering from a knee injury aggravated against usa.both side

23、s boast 100% records heading into the encounter but australia have not conceded a try since josh charnleys effort in their first pool match against england on the opening day.aussie winger jarryd hayne is the competitions top try scorer with nine, closely followed by tuivasa-sheck with eight.but it

24、is recently named rugby league international federation player of the year sonny bill williams who has attracted the most interest in the tournament so far.the kiwi - with a tournament high 17 offloads - has the chance of becoming the first player to win the world cup in both rugby league and rugby

25、union after triumphing with the all blacks in 2011.id give every award back in a heartbeat just to get across the line this weekend, said williams.the (lack of) air up there watch mcayman islands-based webb, the head of fifas anti-racism taskforce, is in london for the football associations 150th an

26、niversary celebrations and will attend citys premier league match at chelsea on sunday.i am going to be at the match tomorrow and i have asked to meet yaya toure, he told bbc sport.for me its about how he felt and i would like to speak to him first to find out what his experience was.uefa hasopened

27、disciplinary proceedings against cskafor the racist behaviour of their fans duringcitys 2-1 win.michel platini, president of european footballs governing body, has also ordered an immediate investigation into the referees actions.cska said they were surprised and disappointed by toures complaint. in

28、 a statement the russian side added: we found no racist insults from fans of cska. baumgartner the disappointing news: mission aborted.the supersonic descent could happen as early as sunda.the weather plays an important role in this mission. starting at the ground, conditions have to be very calm -

29、winds less than 2 mph, with no precipitation or humidity and limited cloud cover. the balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. it will climb higher than the tip of mount everest (5.5 miles/8.85 kilometer

30、s), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. as he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.the balloon will slowly drift to the edge of space at 120,000 feet ( then, i wo

31、uld assume, he will slowly step out onto something resembling an olympic diving platform.they blew it in 2008 when they got caught cold in the final and they will not make the same mistake against the kiwis in manchester.five years ago they cruised through to the final and so far history has repeated itself here - the last try they conceded was scored by englands josh charnley in the ope