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【苏教版】【步步高】2014届高考数学一轮复习备考练习:3.2.2 对数函数(一)]

3.2.2
一、基础过关

对数函数(一)

1.函数 f(x)= 1-2log6x的定义域为________. 2.函数 y=3x(-1≤x<0)的反函数为________. 3. 已知函数 y=ex 的图象与函数 y=f(x)的图象关于直线 y=x 对称, 则 f(2x)=________________. 1 4. 设集合 M={y|y=( )x, x∈[0, +∞)}, N={y|y=log2x, x∈(0,1]}, 则集合 M∪N=________. 2 5.如果函数 f(x)=(3-a)x,g(x)=logax 的增减性相同,则 a 的取值范围是________. 6.已知函数 y=loga(x-3)-1 的图象恒过定点 P,则点 P 的坐标是________. 7.求下列函数的定义域与值域: (1)y=log2(x-2); (2)y=log4(x2+8). 8.设函数 f(x)=ln(x2+ax+1)的定义域为 A. (1)若 1∈A,-3D∈/A,求实数 a 的范围; (2)若函数 y=f(x)的定义域为 R,求实数 a 的取值范围. 二、能力提升 1 9.已知 x=ln π,y=log52,z=e- ,则 x,y,z 的大小关系为________. 2 2 10.若 loga <1,则 a 的取值范围是____________. 3 11.函数 f(x)=log3(2x2-8x+m)的定义域为 R,则 m 的取值范围是________. 12.已知函数 f(x)=loga(1+x),g(x)=loga(1-x),a>0,且 a≠1. (1)设 a=2,函数 f(x)的定义域为[3,63],求函数 f(x)的最值; (2)求使 f(x)-g(x)>0 的 x 的取值范围. 三、探究与拓展 1 13.若不等式 x2-logmx<0 在(0, )内恒成立,求实数 m 的取值范围. 2

答案
1.(0, 6] 1 2.y=log3x( ≤x<1) 3 3.ln 2+ln x(x>0) 4.(-∞,1] 5.(1,2) 6.(4,-1) 7.解 (1)由 x-2>0,得 x>2, 所以函数 y=log2(x-2)的定义域是(2,+∞),值域是 R. (2)因为对任意实数 x,log4(x2+8)都有意义, 所以函数 y=log4(x2+8)的定义域是 R. 又因为 x2+8≥8, 3 所以 log4(x2+8)≥log48= , 2 3 即函数 y=log4(x2+8)的值域是[ ,+∞). 2
? ?1+a+1>0 10 10 ,+∞?. 8.解 (1)由题意,得? ,所以 a≥ .故实数 a 的范围为? 3 ? ? 3 ?9-3a+1≤0 ?

(2)由题意,得 x2+ax+1>0 在 R 上恒成立,则 Δ=a2-4<0,解得-2<a<2. 故实数 a 的范围为(-2,2). 9.y<z<x 解析 ∵x=ln π>ln e,∴x>1. 1 ∵y=log52<log5 5,∴0<y< . 2 1 1 1 1 1 ∴z=e- = > = ,∴ <z<1. 2 2 e 4 2 综上可得,y<z<x. 2 10.(0, )∪(1,+∞) 3 2 解析 由 loga <1 得: 3 2 loga <logaa. 3 2 当 a>1 时,有 a> ,即 a>1; 3 2 2 当 0<a<1 时,则有 a< ,即 0<a< . 3 3

2 综上可知,a 的取值范围是(0, )∪(1,+∞). 3 11.(8,+∞) 12.解 (1)当 a=2 时,函数 f(x)=log2(x+1)为[3,63]上的增函数, 故 f(x)max=f(63)=log2(63+1)=6, f(x)min=f(3)=log2(3+1)=2. (2)f(x)-g(x)>0,即 loga(1+x)>loga(1-x), ①当 a>1 时,1+x>1-x>0,得 0<x<1. ②当 0<a<1 时,0<1+x<1-x,得-1<x<0. 13.解 由 x2-logmx<0,得 x2<logmx, 1 1 要使 x2<logmx 在(0, )内恒成立,只要 y=logmx 在(0, )内的图象在 y=x2 的上方,于是 2 2 0<m<1. 在同一坐标系中作 y=x2 和 y=logmx 的草图,如图所示.

1 1 ∵x= 时,y=x2= , 2 4 1 1 1 1 ∴只要 x= 时,y=logm ≥ =logmm . 2 2 4 4 1 1 1 ∴ ≤m ,即 ≤m.又 0<m<1, 2 4 16 ∴ 1 1 ≤m<1,即实数 m 的取值范围是[ ,1). 16 16

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