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对数第二课时 对数的运算性质及换底公式 课件(29张) 高中数学 必修1 苏教版_图文

第3章 指数函数、对数函数和幂函数 第二课时 对数的运算性质及 换底公式 第3章 指数函数、对数函数和幂函数 学习导航 1.了解对数的换底公式. 学习 2.理解对数的运算性质.(重点) 目标 3.掌握用对数的运算性质进行化简与证明.(重 点、难点). 通过推导对数的运算性质,准确地运用对数运算 学法 性质进行运算、求值、化简,并掌握化简求值的 指导 技能,培养分析、综合解决问题的能力,培养数 学应用的意识和科学分析问题的精神和态度. 1.对数的运算性质 如果a>0,且a≠1,M>0,N>0,那么 logaM+logaN ; (1)loga(MN)=______________ M N (2)Loga logaM-logaN ; =______________ nlogaM (3)logaMn=____________ (n∈R). 2.换底公式 logcN logca = (a>0 且 a≠1 , c>0 且 c≠1 , 一 般 地 , 称 logaN________ N>0)为对数的换底公式. -1 . 1.log23-log26=________ 3 1 - 解析:原式= log2( )=log2 =log22 1=-log22=-1. 6 2 2 2.log23· log34=________. lg 3 lg 4 lg 4 2lg 2 解析:log23·log34= · = = =2. lg 2 lg 3 lg 2 lg 2 3. (课本改编题)写出下列各式的值: 2 (1)log212- log23= ________ ; 2 (2)lg 5+lg20= ________ ; 1 0 (3)log53+ log5 = ________ ; 3 -1 (4)log35- log315= ________. 12 解析: (1)log212- log23= log2 = log24= 2; 3 (2)lg 5+ lg 20= lg(5× 20)= lg 100=2; 1 1 (3)log53+ log5 = log5(3× )= log51= 0; 3 3 5 1 (4)log35- log315= log3 = log3 =- 1. 15 3 10 4.(课本改编题)log89· log2732=________. 9 log29 log232 2log23 5 10 解析:log89·log2732= · = · = . log28 log227 3 3log23 9 对数的运算性质及应用 (Ⅰ )已知 lg 2= 0.301 0, lg 3= 0.477 1,计算下列 对数式: 3 (1)lg 12; (2)lg ; (3)lg 2.4; (4)lg 6. 2 (链接教材 P76 例 5) 7 (Ⅱ )计算:(1)log535- 2log5 + log57- log51.8; 3 (2)2(lg 2)2+ lg 2· lg 5+ ( lg 2) 2- lg 2+ 1; (3)(lg 5)2+ lg 2· lg 50. (链接教材 P76 例 4) [解 ] (Ⅰ )(1)lg 12= lg(3×22)= lg 3+ lg 22= lg 3+2lg 2= 0.477 1+2×0.301 0= 1.079 1. 3 (2)lg = lg 3- lg 2=0.477 1- 0.301 0= 0.176 1. 2 24 (3)lg 2.4= lg = lg 24- lg 10= lg(3×23)- 1= lg 3+3lg 2 10 - 1= 0.477 1+3×0.301 0-1= 0.380 1. 1 1 1 (4)lg 6= lg 6 = (lg 3 + lg 2)= (0.477 1 + 0.301 0)= 2 2 2 0.389 05. 9 (Ⅱ )(1)原式= log5(5× 7)-2(log57- log53)+ log57- log5 5 = log55+ log57- 2log57+ 2log53+ log57-2log53+ log55 = 2log55=2. (2)原式= lg = lg 2(2lg 2+ lg 5)+ ( lg 2= lg 2- 1) 2 2= 1. 2(lg 2+ lg 5)+1- lg 2+ 1- lg (3)原式= (lg 5)2+ lg 2· (lg 2+ 2lg 5) = (lg 5)2+ 2lg 5· lg 2+ (lg 2)2= (lg 5+ lg 2)2= 1. 方法归纳 (1)利用对数的运算性质时,要注意公式成立的前提条件: a>0,a≠1,M >0,N>0,n∈R. (2)要把握住运算性质的本质特征,防止应用时出现错误. (3)利用对数的运算性质,可以把乘、除、乘方运算转化为加 、减、乘的运算.运用此性质,可加快计算速度. (4)对数的化简求值一般是正用或逆用公式,对真数进行处理, 选哪种策略化简,取决于问题的实际情况,一般本着便于真 数化简的原则进行. 1.计算下列各式: 1 - (1) lg 25+ lg 2+ lg 10+ lg(0.01) 1; 2 32 (2)2log32- log3 + log38-3log55. 9 解:(1)原式= lg[25 × 2× 10 × (10 2) 1] - - 1 2 1 2 = lg(5×2× 10 ×102) 7 = lg 10 = . 2 (2)原式= log322+ log3(32×2 5)+ log323-3 - 1 2 7 2 = log3(22×32× 2 5× 23)- 3 - = log332-3 = 2- 3=- 1. 换底公式的应用 已知log189=a,18b=5,试用a和b表示log3645. (链接教材P76练习T4) [解 ] 法一:∵ 18b= 5,∴ log185= b

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