指数与对数函数幂函数知识点总结

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a x n =，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00=n 。

当n 是奇数时，a a n

n =，

当n 是偶数时，???<≥-==)

0()0(||a a a a a a n n 2．分数指数幂

正数的分数指数幂的意义，规定：

:

)

1,,,0(*>∈>=n N n m a a a

n m n

m )1,,,0(1

1

*>∈>=

=

-

n N n m a a a

a

n

m

n

m n

m

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

（1）r a ·s

r r a a += ),,0(R s r a ∈>；

（2）rs s r a a =)( ),,0(R s r a ∈>； （3）

s r r a a ab =)( ),,0(R s r a ∈>．

（二）指数函数及其性质 <

1、指数函数的概念：一般地，函数)1,0(≠>=a a a y x 且叫做指数函数，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．

注意：利用函数的单调性，结合图象还可以看出： （1）在[a ，b]上，)1a 0a (a )x (f x ≠>=且值域是)]b (f ),a (f [或)]a (f ),b (f [；

（2）若0x ≠，则1)x (f ≠；)x (f 取遍所有正数当且仅当R x ∈； ！

（3）对于指数函数)1a 0a (a )x (f x ≠>=且，总有a )1(f =；

二、对数函数 （一）对数

1．对数的概念：一般地，如果N a x =)1,0(≠>a a ，那么数x 叫做以．a 为．底．N 的对数，记作：N x a log =（a — 底数，N — 真数，N a log — 对数式）

说明：○1 注意底数的限制0>a ，且1≠a ；

○2 x N N a a x =?=log ； ` ○

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N lg ； ○

2 自然对数：以无理数 71828.2=e 为底的对数的对数N ln ．

指数式与对数式的互化

幂值 真数

.

b

对数 （二）对数的运算性质

如果0>a ，且1≠a ，0>M ，0>N ，那么： 。

○

1 M a (log ·=)N M a log ＋N a log ； ○

2 =N

M

a log M a log －N a log ； ○

3 n a M log n =M a log )(R n ∈． 注意：换底公式

a

b

b c c a log log log = （0>a ，且1≠a ；0>c ，且1≠c ；0>b ）．

利用换底公式推导下面的结论

（1）b m

n

b a n a m

log log =；（2）a b b a log 1log =．

/

（二）对数函数

1、对数函数的概念：函数0(log >=a x y a ，且)1≠a 叫做对数函数，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：x y 2log 2=，5

log 5x y = 都不是对数函数，

而只能称其为对数型函数．

○

2 对数函数对底数的限制：0(>a ，且)1≠a ．

幂 函 数

一般地，形如)R a (x y a ∈=的函数称为幂函数，其中a 为常数。

幂函数中，当12

1

321a -=，，，，时性质如下表所示：画图

函数 特征 性质 y=x y x =2

y x =3

y x

=

12

y x =-1 定义域 R R R [0，+∞） {|}x x ≠0

值域 R [0，+∞）

R [0，+∞）

{|}y y ≠0

x ∈+∞[)0，增 x ∈+∞()0，增 单调性 增 x ∈-∞(]，0减 增 增 x ∈-∞()，0减

所过定点

（1，1） （0，0）

（1，1） （0，0）

（1，1） （0，0）

（1，1） （0，0）

（1，1）

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；

（3）如果a>0，则幂函数的图象通过原点，并且在区间)0[∞+，上是增函数；

（4）如果a<0，则幂函数在区间()0，+∞上是减函

Not surprisingly, many aspects of people's daily lives have undergone considerable changes because of the recent development in technology. It is a particular concern that the pace of everyday life is becoming faster, resulting from the development in cars, air travel, telecommunications technology and the Internet 。 As well as benefits, this trend is to bring problems.

On the positive side, the fast rhythm of life requires people to enhance efficiency when working and then allows them to enjoy longer leisure time. With the advance in technology people can make inquires by phone, instead of travelling long distances, Internet access makes it possible (for one to perform various tasks without leaving their offices. Even though people have to travel every now and then, for meeting business partners, visiting clients in other cities or other purposes, modem transport networks reduce the amount of time they spend on commutes.

The acceleration of the pace of life also implies the expansion of people's social circle. In the past, social relationships were limited by physical factors such as geographical distance and low mobility, but nowadays, one can travel further and get acquainted with more people with those technological advances, such as the railroad, the automobile and the telephone. For instance, the rapid penetration of telecommunications technology has made the mobile phone a key social tool and people rely on their mobile phone address book to keep in touch with their friends. On the negative side, the fast-paced lifestyle is responsible for the upsurge in lifestyle-related problems. Jobs become demanding and require workers' full commitment, resulting in their depression and pressure. Underneath the facade of continued contraction of official working hours, employees are actually working longer, primarily because fax, e-mail or other communication devices have made them

accessible to their supervisors, colleagues and customers after work. They have to respond instantly to voice and email messages from others. Private life has to be sacrificed.

According to the facts outlined above, the doubts about the negative effects of the acceleration of pace of life are not well-grounded. People now enjoy greater well-being, which is reflected in more quality family time, less travel-related stress and close contact with friends and family members. However, they might have to accept frequent intrusions as a by-product of convenient communication.