指数与对数函数幂函数知识点总结

发布时间:2024-11-25

指数函数与对数函数知识点总结

(一)指数与指数幂的运算

1.根式的概念:一般地,如果xn a,那么x叫做a的n次方根,其中n>1,且n∈N*.

负数没有偶次方根;0的任何次方根都是0,记作0 0。

a(a 0)当n当n an |a| an a,

a(a 0)

2.分数指数幂

正数的分数指数幂的意义,规定:

a am(a 0,m,n N*,n 1)a

mn

mn

1

mn

1

a

0的正分数指数幂等于0,0的负分数指数幂没有意义 3.实数指数幂的运算性质

rrr s

(1)a〃a a (a 0,r,s R);

rsrs(a) a(2) (a 0,r,s R);

am

(a 0,m,n N*,n 1)

(a 0,r,s R). (3)(ab) aa

(二)指数函数及其性质

1、指数函数的概念:一般地,函数y ax(a 0,且a 1)叫做指数函数,其中x是自变量,函数的定义域为R.

注意:指数函数的底数的取值范围,底数不能是负数、零和1.

rrs

注意:利用函数的单调性,结合图象还可以看出:

(1)在[a,b]上,f(x) ax(a 0且a 1)值域是[f(a),f(b)]或[f(b),f(a)];

(2)若x 0,则f(x) 1;f(x)取遍所有正数当且仅当x R;

(3)对于指数函数f(x) ax(a 0且a 1),总有f(1) a; 二、对数函数 (一)对数

1.对数的概念:一般地,如果ax N(a 0,a 1),那么数x叫做以.a为底..N的对数,记作:x logaN(a— 底数,N— 真数,logaN— 对数式) 说明:○1 注意底数的限制a 0,且a 1;

2 ax N logaN x; ○

3 注意对数的书写格式. ○

两个重要对数:

1 常用对数:以10为底的对数lgN; ○

2 自然对数:以无理数e 2.71828 为底的对数的对数○

lnN.

指数式与对数式的互化

幂值 真数

对数 (二)对数的运算性质

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). ○

注意:换底公式

logcb

(a 0,且a 1;c 0,且c 1;b 0). logab

logca

利用换底公式推导下面的结论

1n

(1)logabn logab;(2)logab .

mlogba

(二)对数函数

1、对数函数的概念:函数y logax(a 0,且a 1)叫做对

m

数函数,其中x是自变量,函数的定义域是(0,+∞). 注意:○1 对数函数的定义与指数函数类似,都是形式定义,注意辨别。如:y 2log2x,y log5x 都不是对数函数,

5

而只能称其为对数型函数.

2 对数函数对底数的限制:(a 0,且a 1).

幂 函 数

一般地,形如y xa(a R)的函数称为幂函数,其中a为常数。

1,

2,3, 1时性质如下表所示:画图 幂函数中,当a 1

2

结合以上特征,得幂函数的性质如下:

(1)所有的幂函数在(0, )都有定义,并且图象都通过点(1,1); (2)当a为奇数时,幂函数为奇函数;当a为偶数时,幂函数为偶函数;

)上是增函数; (3)如果a>0,则幂函数的图象通过原点,并且在区间[0,(4)如果a<0,则幂函数在区间(0, )上是减函

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