对数与对数函数讲义.doc

对数与对数函数（一） 对数1对数的概念：一般地，如果，那么数叫做以为底的对数，记作：（ 底数， 真数， 对数式）说明： 注意底数的限制，且； ； 注意对数的书写格式两个重要对数： 常用对数：以10为底的对数； 自然对数：以无理数为底的对数的对数指数式与对数式的互化 幂值 真数 N b 底数 指数 对数（二）对数的运算性质如果，且，那么： ·； ； 注意：换底公式（，且；，且；）利用换底公式推导下面的结论（1）；（2）（二）对数函数1、对数函数的概念：函数，且叫做对数函数，其中是自变量，函数的定义域是（0，+）注意： 对数函数的定义与指数函数类似，都是形式定义，注意辨别。如：， 都不是对数函数，而只能称其为对数型函数 对数函数对底数的限制：，且2、对数函数的性质：a>10<a<1定义域x0定义域x0值域为R值域为R在R上递增在R上递减函数图象都过定点（1，0）函数图象都过定点（1，0）题型一：对数的定义与对数运算【例1】 将下列指数式化为对数式，对数式化为指数式：；.求下列各式中的值：；.【例2】 将下列指数式化为对数式，对数式化为指数式：（1）； （2）； （3）；（4）； （5）； （6）ln100=4.606.【例3】 将下列对数式写成指数式：（1）；（2）；（3）； （4）【例4】 已知, 则的值等于（ ）.A. 1 B. 2 C. 8 D. 12【例5】 计算下列各式的值：（1）； （2）； （3）.【例6】 ，【例7】 （）等于（ ）.A. 1 B. 1C. 2D. 2【例8】 （a0）化简得结果是（）.A. aB. a2C. a D. a【例9】 化简的结果是（ ）. A. B. 1 C. 2 D.【例10】 计算 . 【例11】 计算：=_【例12】 化简与求值：；【例13】 若，则= . 【例14】 化简的结果是 （ ）. A .1 B. C. 2 D.3 【例15】 计算： 5 【例16】 求下列各值：；.【例17】 设，求实数m的值【例18】 （1）设，求的值.（2）设,，且，求a的值.题型二：对数运算法则的应用【例19】 若、，且、，则A.B.C.或D.、为一切非1的正数【例20】 下列各式中，正确的是（ ）A.B.C.D.【例21】 已知求证：【例22】 已知，用 a 表示【例23】 若，则 . 【例24】 已知，用表示【例25】 已知且，则等于A.B.C.D.【例26】 已知，用表示.【例27】 （1）已知，试用a、b表示的值；（2）已知，用a、b表示. 【例28】 已知，求题型三：对数方程【例29】 求底数：（1）, （2）【例30】 已知，求实数的值.【例31】 证明：【例32】 求x的值： 【例33】 解方程【例34】 方程的解x= ；【例35】 设a为实常数，解关于x的方程.【例36】 设正数a，b，c满足.（1）求证：；（2）又设，求a，b，c的值.题型一 对数函数的基本性质【例1】 下面结论中，不正确的是 A.若a1，则与在定义域内均为增函数B.函数与图象关于直线对称C.与表示同一函数D.若，则一定有【例2】 图中的曲线是的图象，已知的值为，则相应曲线的依次为（ ）. A. ， B. ， C. ， D. ，0xC1C2C4C31y【例3】 当时,在同一坐标系中,函数的图象是（ ）. xy11oxyo11oyx11oyx11 A B C D【例4】 设，函数在区间上的最大值与最小值之差为，则（ ）. A.B. 2C. D. 4【例5】 若，则a的取值范围是 A.B.C.D.或a1【例6】 比较两个对数值的大小： ； .【例7】 若，那么满足的条件是（ ）. A. B. C. D. 【例8】 已知，则（）A.B.C.D.【例9】 下列各式错误的是（ ）. A. B. C. D. .【例10】 若，则（）A.B.C.D.【例11】 若，求的关系。【例12】 已知，求的取值范围.题型二 对数型符合型复合函数的定义域值域【例13】 下列函数中哪个与函数y=x是同一个函数（ ） A. B. y= C. D. y=【例14】 函数的定义域是（ ）. A. B. C. D. 【例15】 函数的定义域为 . （用区间表示）【例16】 求下列函数的定义域：（1） (2)【例17】 求下列函数的定义域：；.【例18】 求下列函数的定义域：；.【例19】 求下列函数的定义域： 【例20】 求下列函数的定义域：（1） ； （2）.【例21】 函数的值域是（ ）. A. R B. C. D. 【例22】 函数的值域是 A.y0B.yRC.y0且y1D.y2待添加的隐藏文字内容2【例23】 求下列函数的定义域、值域：1234题型三 对数型符合型复合函数的单调性【例24】 下列函数中，在上为增函数的是（ ）. A. B. C. D. 【例25】 证明函数y= (+1)在（0，+）上是减函数；【例26】 讨论函数的单调性.【例27】 求的单调递减区间【例28】 求函数的单调递增区间【例29】 求函数的单调区间【例30】 求函数的单调区间题型四 对数函数的综合与应用【例31】 函数是 函数. （填“奇”、“偶”或“非奇非偶”）【例32】 如果，求的取值范围.【例33】 已知，要使AB，求实数k的取值范围.【例34】 已知函数，求：（1）的值域； （2）的最大值及相应x的值.【例35】 已知函数其中（1）求函数的定义域； （2）判断的奇偶性，并说明理由；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 injury, 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 Charnley's effort in their first pool match against England on the opening day.Aussie winger Jarryd Hayne is the competition's 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 tournament 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."I'd 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 Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's 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 it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's 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 as 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 troposphere) 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 tropopause),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 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 England's Josh Charnley in the opening game of the tournament.That could be classed as a weakness, a team under-cooked - but 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. They've been great at all the unglamorous stuff that 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 have 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 Cup, 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 sides boast 100% records heading into the encounter but Australia have not conceded a try since Josh Charnley's effort in their first pool match against England on the opening day.Aussie winger Jarryd Hayne is the competition's 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 tournament 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."I'd 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 Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's 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 it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's 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 as 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 troposphere) 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 tropopause),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 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 England's Josh Charnley in the opening game of the tournament.That could be classed as a weakness, a team under-cooked - but 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. They've been great at all the unglamorous stuff that 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 have 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 Cup, and know they can rise to the big occasion.【例36】