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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