人教版高中數(shù)學(xué)必修一第二章　基本初等函數(shù)第二節(jié)《對(duì)數(shù)的概念》參考課課件

1、你能用等式表示數(shù)字你能用等式表示數(shù)字9 9，3 3，2 2之間的關(guān)系嗎？之間的關(guān)系嗎？我發(fā)現(xiàn)我發(fā)現(xiàn)9=39=32 2我發(fā)現(xiàn)我發(fā)現(xiàn)3=3= 9 92=2=？ 一般地，如果一般地，如果ax=N(=N(a00，且，且a1)1)，那么數(shù)，那么數(shù)x x叫做以叫做以a a為底為底N N的對(duì)數(shù)的對(duì)數(shù)(logarithm)(logarithm)，記作：，記作： 其中其中a a叫做對(duì)數(shù)的底數(shù)，叫做對(duì)數(shù)的底數(shù)，N N叫做真數(shù)叫做真數(shù)。 ax=N =N logloga aN N 式子式子 名名 稱稱 指數(shù)式指數(shù)式 對(duì)數(shù)式對(duì)數(shù)式 指數(shù)指數(shù) 真數(shù)真數(shù) =N =N =b =b a a b b N N logloga aN

2、 N 對(duì)數(shù)式與指數(shù)式的對(duì)比對(duì)數(shù)式與指數(shù)式的對(duì)比: :a ax x 底數(shù)底數(shù) 對(duì)數(shù)對(duì)數(shù) 冪值冪值 底數(shù)底數(shù) 用連線表示下列兩式中字母的對(duì)應(yīng)關(guān)系：用連線表示下列兩式中字母的對(duì)應(yīng)關(guān)系：a ab b=N log=N loga aN=b N=b 式子式子 取值范圍取值范圍 指數(shù)式指數(shù)式 對(duì)數(shù)式對(duì)數(shù)式 =N =N =b =b a a b b N N bR bR logloga aN N 為什么在對(duì)數(shù)中要規(guī)定為什么在對(duì)數(shù)中要規(guī)定a a0 0，且，且a1a1？a ax x a0a0，且，且 a1 a1 N0 N0 a0a0，且，且 a1 a1 bR bR N0 N0 通常我們將以通常我們將以1010為底的對(duì)數(shù)

3、叫做為底的對(duì)數(shù)叫做常用對(duì)數(shù)常用對(duì)數(shù)(common logarithm)(common logarithm)，并把，并把loglog1010N N記為記為: : 在科學(xué)技術(shù)中常使用以無理數(shù)在科學(xué)技術(shù)中常使用以無理數(shù)e=2.71828e=2.71828為底數(shù)的對(duì)數(shù)，以為底數(shù)的對(duì)數(shù)，以e e為底的對(duì)數(shù)稱為為底的對(duì)數(shù)稱為自然對(duì)數(shù)自然對(duì)數(shù)(natural logarithm)(natural logarithm)，并且把，并且把logloge eN N記為記為: : lgN lgN lnN lnN 討論：討論：你能用對(duì)數(shù)表示你能用對(duì)數(shù)表示2 2x x=-3=-3和和2 2x x=0=0嗎？為什么？嗎？為

4、什么？ (1) (1) 負(fù)數(shù)和零沒有對(duì)數(shù)；負(fù)數(shù)和零沒有對(duì)數(shù)；在中，必須在中，必須logloga aN0N0，這是由于在實(shí)數(shù)范圍，這是由于在實(shí)數(shù)范圍內(nèi)，正數(shù)的任何次冪都是正數(shù)，因而內(nèi)，正數(shù)的任何次冪都是正數(shù)，因而a ax x=N=N中中N N總總是正數(shù)。是正數(shù)。(2) (2) 對(duì)任意的對(duì)任意的a a0 0且且a1a1，都有，都有a a0 0=1=1。 所以所以logloga a1=1=0 0(3) (3) 對(duì)任意的對(duì)任意的a a0 0且且a1a1，都有，都有a a1 1=a=a。 所以所以logloga aa=a=1 1 將下列指數(shù)式化為對(duì)數(shù)式將下列指數(shù)式化為對(duì)數(shù)式 (1)5(1)54 4 =

5、625 = 625 (2)2(2)2-6 -6 = = 1 12 2m m (3)(3)1 164 64 = 5.73 = 5.73 loglog5 5625 = 4 625 = 4 loglog5 5 = -6 = -6 1 164 64 loglog 5.73 = m 5.73 = m 1 12 2將下列對(duì)數(shù)式化為指數(shù)式將下列對(duì)數(shù)式化為指數(shù)式1010-2-2 = 0.01 = 0.01 (1)(1)(2)lg0.01(2)lg0.01 = -2 = -2 (3)(3)ln10 = 2.303 ln10 = 2.303 loglog 16 = -4 16 = -4 1 12 21 12 2-

6、4 -4 = 16 = 16 e e2.303 2.303 = 10 = 10 求下列各式中的求下列各式中的x x值值(1)log(1)log64648 = 8 = 2 23 3- -(2)log(2)logx x8 = 6 8 = 6 (3)lg100 = x (3)lg100 = x (4)-lne(4)-lne2 2 = x = x 解：解：(1)(1)因?yàn)橐驗(yàn)閘oglog64648 = 8 = 2 23 3- -所以所以64 64 2 23 3- -= (4= (43 3) )2 23 3- -= 4= 4-2-21 116 16 = =(2)(2)因?yàn)橐驗(yàn)閘oglogx x8 = 6

7、, 8 = 6, 所以所以x x6 6 = 8= 8， 又又x x0 0 x = 8 x = 8 1 16 6= (2= (23 3) )1 16 6= 2= 21 12 2= =2 2求下列各式中的求下列各式中的x x值值(1)log(1)log64648 = 8 = 2 23 3- -(2)log(2)logx x8 = 6 8 = 6 (3)lg100 = x (3)lg100 = x (4)-lne(4)-lne2 2 = x = x 解：解：(3)(3)因?yàn)橐驗(yàn)閘g100 = xlg100 = x，所以，所以1010 x x = 100 = 100 1010 x x = 10= 10

8、2 2， 于是于是x = 2 x = 2 (4)(4)因?yàn)橐驗(yàn)?lne-lne2 2 = x, = x, 所以所以lnelne2 2 = -x= -x，e e2 2 = e= e-x-x， 于是于是x = -2 x = -2 將下列指數(shù)式寫成對(duì)數(shù)式將下列指數(shù)式寫成對(duì)數(shù)式(1) 2(1) 23 3 = 8 (2)27 = = 8 (2)27 =1 13 3- -1 13 3(3)10(3)10 x x = 25= 25解：解：(1)3 = log(1)3 = log2 28 8 (3)x = log(3)x = log10102525(2) = log(2) = log27 27 1 13 31

9、 13 3- -指數(shù)式化為對(duì)數(shù)式：指數(shù)式化為對(duì)數(shù)式：冪的底數(shù)變?yōu)閷?duì)數(shù)函數(shù)的冪的底數(shù)變?yōu)閷?duì)數(shù)函數(shù)的底數(shù)，指數(shù)變對(duì)數(shù)，冪值底數(shù)，指數(shù)變對(duì)數(shù)，冪值變真數(shù)。變真數(shù)。 將下列對(duì)數(shù)式寫成指數(shù)式將下列對(duì)數(shù)式寫成指數(shù)式解：解： (1)5(1)5x x = 27= 271 13 3(2)7(2)7x x = =(3)10(3)10 x x = 0.3= 0.3(4)e(4)ex x = =3 3(1)x = log(1)x = log5 527 27 (2)x = log(2)x = log7 7 (3)x = lg0.3 (3)x = lg0.3 (4)x = ln (4)x = ln 3 31 13 3將下

10、列指數(shù)式寫成對(duì)數(shù)式將下列指數(shù)式寫成對(duì)數(shù)式解：解：(1)(1)設(shè)設(shè)x = logx = log2 22525， (1)log(1)log5 525 25 (3)ln(3)ln(2)log(2)log2 2 1 116 16 e e(4)lg0.001 (4)lg0.001 (5)log(5)log151515 15 (6)log(6)log0.40.41 1 則則5 5x x = 25 = 25 = 5= 52 2所以所以x = 2 x = 2 (2)(2)設(shè)設(shè)x = logx = log2 2 1 116 16 , ,則則2 2x x = = 1 116 16 = 2= 2-4-4所以所以x

11、= -4 x = -4 (3)(3)設(shè)設(shè)x = ln x = ln e e , , 則則e ex x = = e e = e= e1 12 2所以所以x = x = 1 12 2將下列指數(shù)式寫成對(duì)數(shù)式將下列指數(shù)式寫成對(duì)數(shù)式解：解：則則1010 x x = 0.001 = 0.001 = 10= 10-3-3所以所以x = -3 x = -3 (5)(5)設(shè)設(shè)x = logx = log151515,15, 則則x = 1 x = 1 (6)(6)設(shè)設(shè)x = logx = log0.40.41, 1, 則則x = 0 x = 0(4)(4)設(shè)設(shè)x = lg0.001x = lg0.001，(1)

12、log(1)log5 525 25 (3)ln(3)ln(2)log(2)log2 2 1 116 16 e e(4)lg0.001 (4)lg0.001 (5)log(5)log151515 15 (6)log(6)log0.40.41 1 若若a ax x=N(a0=N(a0，且，且a1)a1)，那么數(shù)，那么數(shù)x x叫做以叫做以a a為為底底N N的對(duì)數(shù)的對(duì)數(shù)(logarithm)(logarithm)，記當(dāng)，記當(dāng)x=logx=loga aN N，當(dāng)，當(dāng)a=10a=10時(shí)稱作常用對(duì)數(shù)，而時(shí)稱作常用對(duì)數(shù)，而a=ea=e時(shí)，則稱自然對(duì)數(shù)。時(shí)，則稱自然對(duì)數(shù)。 1616世紀(jì)末至世紀(jì)末至1717世紀(jì)初的時(shí)候，當(dāng)時(shí)在自然世紀(jì)初的時(shí)候，當(dāng)時(shí)在自然科學(xué)領(lǐng)域科學(xué)領(lǐng)域( (特別是天文學(xué)特別是天文學(xué)) )的發(fā)展上經(jīng)常遇到大的發(fā)展上經(jīng)常遇到大量精密而又龐大的數(shù)值計(jì)算，于是數(shù)學(xué)家們?yōu)榱烤芏铸嫶蟮臄?shù)值計(jì)算，于是數(shù)學(xué)家們?yōu)榱藢で蠡?jiǎn)的計(jì)算方法而發(fā)明了對(duì)數(shù)。了尋求化簡(jiǎn)的計(jì)算方法而發(fā)明了對(duì)數(shù)。 1624 1624年，英國的布里格斯創(chuàng)造了常用對(duì)數(shù)。年，英國的布里格斯創(chuàng)造了常用對(duì)數(shù)。 1619 1619年，倫敦斯彼得所著的年，倫敦斯彼得所著的新對(duì)數(shù)新對(duì)數(shù)使對(duì)使對(duì) 對(duì)數(shù)與自然對(duì)數(shù)更接近（以對(duì)數(shù)與