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高中数学苏教版必修1 3.2.1第二课时 对数的运算性质及换底公式 作业 含解析

[学业水平训练] 一、填空题 2 1.给出下列 4 个等式:①log253=3log25;②log253=5log23;③log84= ; 3 ④log 24=4.其中正确的等式是________.(写出所有正确的序号) 1 解析:②中 log253= log23,故②不正确,①③④都正确. 5 答案:①③④ 2.2log510+log50.25 的值等于________. 解析:原式=log5102+log50.25=log5(102×0.25)=log525=2. 答案:2 log2716 3. 的值为________. log34 lg 16 解析:原式= lg 27 lg 16·lg 3 2lg 4·lg 3 2 = = = . lg 4 lg 27·lg 4 3lg 3·lg 4 3 lg 3 答案: 2 3 4.若 log34·log48·log8m=log416,则 m=________. lg 4 lg 8 lg m 解析:由已知,得 log34·log48·log8m= · · =log3m=2,∴m=32 lg 3 lg 4 lg 8 =9. 答案:9 5.log 1 1 +( )log2 =________. 2 2 4 2( 2 2 解析:原式=log 答案:3 1 1 )+2-log2 =-1+4=3. 4 2 1 1 6.已知 2m=3n=36,则 + =________. m n 1 1 解析:m=log236,n=log336,∴ =log362, =log363, m n 1 1 1 ∴ + =log366= . m n 2 答案: 1 2 二、解答题 7.(1)已知 log95=a,log37=b,试用 a,b 表示 log2135. (2)已知 log35=m,log83=n,试用 m,n 表示 lg 5. 解:(1)由 log95=a,得 a= log35 1 = log35. log39 2 log335 log35+log37 2a+b ∴log2135= = = . log321 log33+log37 1+b lg 5 ? =m, ? ?log 5=m ?lg 3 (2)由? ?? ? log 3 = n lg 3 ? ? ?3lg 2=n, 3 8 ① ② lg 5 lg 5 ①×②,得 =mn,即 =mn, 3lg 2 3(1-lg 5) 3mn ∴lg 5= . 1+3mn 1 1 1 8.已知 3x=4y=6z,求证: + = . x 2y z 证明:设 3x=4y=6z=t, 则 x=log3t,y=log4t,z=log6t(t>0,t≠1), 1 1 1 1 1 1 1 1 1 所以 + = + =logt3+logt2=logt6= = .∴ + = . x 2y log3t 2log4t log6t z x 2y z [高考水平训练] 一、填空题 1.lg( 3+ 5+ 3- 3+ 5)的值为________. 5+ 3- 5)2 1 解析:原式= lg( 2 1 = lg(6+2 2 1 = . 2 答案: 1 2 4 9-5) 2.计算 log8(log2 2)的值为________. 1 2 2)=log8 =-2log82=- . 4 3 解析:log8(log2 4 答案:- 2 3 二、解答题 1 1 1 3.设 xa=yb=zc,且 + = ,求证:z=xy. a b c 证明:设 xa=yb=zc=k, 1 lg x 1 lg y 1 lg z 则 = , = , = , a lg k b lg k c lg k lg x lg y lg z ∴ + = , lg k lg k lg k ∴lg(xy)=lg z,即 z=xy. 2 4.地震的震级 R 与地震释放的能量 E 的关系为 R= (lg E-11.4).2008 年 5 3 月 12 日,中国汶川发生了 8.0 级特大地震,而 2011 年 3 月 11 日日本海域地 震的震级为 9.0 级,那么 2011 年地震的能量约是 2008 年地震的多少倍. 2 ? 8= (lg E -11.4) ? ? 3 ?lg E =23.4 解:由题意可知:? ?? ? 2 ?lg E =24.9 9= (lg E -11.4) ? ? 3 1 1 2 2 ?lg E2 3 =lg E2-lg E1= . E1 2 3 3 E2 ∴ =102.故为 102倍. E1

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