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24466.com:2018-2019版高中数学苏教版必修一课件:3.2.1 第2课时 对数的运算性质_图文

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第3章 3.2.1 对 数 第2课时 对数的运算性质 学习目标 1.掌握积、商、幂的对数运算性质,理解其推导过程 和成立条件. 2.掌握换底公式及其推论. 3.能熟练运用对数的运算性质进行化简求值. 内容索引 问题导学 题型探究 当堂训练 问题导学 知识点一 对数运算性质 思考 有了乘法口诀,我们就不必把乘法还原成为加法来计算 .那么, 有没有类似乘法口诀的东西,使我们不必把对数式还原成指数 式就能计算? 答案 有 . 例如,设 logaM = m , logaN = n ,则 am = M , an = N , ∴MN=am· an=am+n, ∴loga(MN)=m+n=logaM+logaN. 得到的结论loga(MN)=logaM+logaN可以当公式直接进行对数运算. 答案 梳理 一般地,如果a>0,且a≠1,M>0,N>0,那么 (1)loga(M· N)= logaM+logaN ; M (2)loga = logaM-logaN ; N (3)logaMn= nlogaM (n∈R). 知识点二 换底公式 思考1 观察知识点一的三个公式,我们发现对数都是同底的才能用这 三个公式 .而实际上,早期只有常用对数表 ( 以10为底) 和自然对 数表(以无理数e为底),可以查表求对数值.那么我们在运算和求 值中遇到不同底的对数怎么办? 答案 设法换为同底. 答案 思考2 log25 假设 =x,则log25=xlog23,即log25=log23x,从而有3x=5, log23 再化为对数式可得到什么结论? 答案 把3x=5化为对数式为log35=x, log25 log25 又因为 x= ,所以得出 log35= 的结论. log23 log23 答案 梳理 一般地,我们有logaN= logcN ,其中a>0,a≠1,N>0,c>0,c≠1.这个 logca 公式称为对数的换底公式. 题型探究 类型一 具体数字的化简求值 例1 计算:(1)log345-log35; 解 45 log345-log35=log3 =log39=log332=2log33=2. 5 (2)log2(23×45); 解 log2(23×45)=log2(23×210)=log2(213)=13log22=13. 解答 lg 27+lg 8-lg (3) lg 1.2 解 1 000 ; 3 lg? 27×8?-lg 10 2 原式= 12 lg 10 3 2 3 lg? 3 ×2 ÷1 0 ? = = 12 12 lg 10 lg 10 3 2 ?3×4? ? ? lg? ? ? 10 ? 3 2 3 12 lg 2 10 3 = = . 12 2 lg 10 解答 (4)log29· log38. 解 log29· log38=log2(32)· log3(23) =2log23· 3log32 1 =6· log23· log23 =6. 解答 反思与感悟 具体数的化简求值主要遵循2个原则 (1)把数字化为质因数的幂、积、商的形式. (2)不同底化为同底. 跟踪训练1 计算:(1)2log63+log64; 解 原式=log632+log64=log6(32×4)=log6(62)=2log66=2. 1 1 ? (2)(lg 25-lg 4)÷1 0 0 2 ; 1 25 2? ( ? ) 解 原式=(lg 1 )÷1 0 2 =lg 102÷ 10-1=2×10=20. 4 (3)log43· log98; lg 3 lg 8 lg 3 3lg 2 3 解 原式= · = · = . lg 4 lg 9 2lg 2 2lg 3 4 解答 (4)log2.56.25+ln e- 0 .0 6 4 3 . 解 1 ( 64 ) 1 4 2 原式=log2.5(2.5) +2- =2+2-10 1 000 3 1 1 21 =10. 解答 类型二 代数式的化简 命题角度1 代数式恒等变换 2 x y 例2 化简loga . 3 z 解答 反思与感悟 使用公式要注意成立条件,如lg x2不一定等于2lg x,反例:log10(-10)2 = 2log10( - 10) 是 不 成 立 的 . 要 特 别 注 意 loga(MN)≠logaM· logaN , loga(M±N)≠logaM±logaN. x 跟踪训练 2 已知 y>0,化简 loga yz . x 解 ∵ yz >0,y>0,∴x>0,z>0. x 1 ∴loga yz =loga x-loga(yz)=2 logax-logay-logaz. 解答 命题角度2 用代数式表示对数 例3 已知log189=a,18b=5,求log3645. 解答 反思与感悟 此类问题的本质是把目标分解为基本“粒子”,然后用指定字母换元. 跟踪训练3 已知log23=a,log37=b,用a,b表示log4256. 解 1 ∵log23=a,则a=log32, 又∵log37=b, log37+3log32 ab+3 log356 ∴log4256= = = . log342 log37+log32+1 ab+a+1 解答 当堂训练 1 0 1.log53+log53 等于________. 1 2 3 4 5 答案 2.lg 5+lg 1 20的值是________. 20=lg 100=lg 10=1. 解析 lg 5+lg 1 2 3 4 5 解析 答案 4 3.log29×log34等于________. 1 2 3 4 5 答案 4.lg 0.01+log216的值是________. 2 解析 lg 0.01+log

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