Generally speaking, if the power of a (a is greater than 0, and a is not equal to 1) is equal to n, then this number b is called the logarithm of n with the base of a, and it is recorded as log aN=b, where a is called the base of logarithm and n is called a real number.

If the real number formula has no root number, then as long as the real number formula is greater than zero, if there is a root number, the real number is required to be greater than zero, and the formula in the root number is greater than zero.

Cardinality is greater than 0 instead of 1.

Why is the base of logarithmic function greater than 0 instead of 1?

In the ordinary logarithmic formula, a

The general form of logarithmic function is that it is actually the inverse function of exponential function, which can be expressed as x = a y, so the stipulation of a in exponential function is also applicable to logarithmic function.

The figure on the right shows the function diagram of different size A:

You can see that the graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions.

The domain of (1) logarithmic function is a set of real numbers greater than 0.

(2) The range of logarithmic function is the set of all real numbers.

(3) The function always passes (1, 0).

(4) When a is greater than 1, it is monotone increasing function and convex; When a is less than 1 and greater than 0, the function is monotonically decreasing and concave.

(5) Obviously, the logarithmic function is unbounded.

Operational properties of logarithmic function;

If a > 0 and a is not equal to 1, m >；; 0, N>0, then:

( 1)log(a)(MN)= log(a)(M)+log(a)(N)；

(2)log(a)(M/N)= log(a)(M)-log(a)(N)；

(3) log (a) (m n) = nlog (a) (m) (n belongs to r)

2. Exponential function

The general form of exponential function is y = a x(a >；; 0 and not = 1), from the above discussion of power function, we can know that if X can take the whole set of real numbers as the domain, then we only need to make.

As shown in the figure, different sizes of a will affect the function diagram.

In the function y = a x, you can see:

The domain of (1) exponential function is the set of all real numbers, provided that a is greater than 0 and not equal to 1. For the case that a is not greater than 0, there must be no continuous interval in the definition domain of the function, so we will not consider it.

At the same time, a equal to 0 is generally not considered.

(2) The range of exponential function is a set of real numbers greater than 0.

(3) The function graph is concave.

(4) If a is greater than 1, the exponential function increases monotonically; If a is less than 1 and greater than 0, it is monotonically decreasing.

(5) We can see an obvious law, that is, when a tends to infinity from 0 (of course, it can't be equal to 0), the curves of the functions tend to approach the positions of monotonic decreasing functions of the positive semi-axis of Y axis and the negative semi-axis of X axis respectively. The horizontal straight line y= 1 is the transition position from decreasing to increasing.

(6) Functions always infinitely tend to a certain direction on the X axis and never intersect.

(7) The function always passes (0, 1).

Obviously the exponential function is unbounded.

(9) Exponential function is neither odd function nor even function.

(10) When a in two exponential functions is reciprocal, the functions are even.

Example 1: Are the following functions increasing function or subtraction functions on R? Explain why.

⑴y=4^x

Because 4> 1, y = 4 x is the increasing function on R;

⑵y=( 1/4)^x

Because 0

3. Power function

The general form of power function is y = x a.

It is easy to understand if A takes a nonzero rational number, but it is not easy to understand if A takes an irrational number. In our course, we don't need to master the problem of how to understand exponential irrational numbers, because it involves very advanced knowledge of real number continuum. So we can only accept it as a known fact.

For the value of a nonzero rational number, it is necessary to discuss their respective characteristics in several cases:

Firstly, we know that if a=p/q, and p/q is an irreducible fraction (i.e. p and q are coprime), and both q and p are integers, then x (p/q) = the root of q (the power of x), if q is odd, the domain of the function is r, if q is even, the domain of the function is [0, +∞]. When the exponent n is a negative integer, let a=-k, then x = 1/(x k), obviously x≠0, and the domain of the function is (-∞, 0)∩(0, +∞). So it can be seen that the limitation of X comes from two points. First, it can be used as a denominator, but it cannot be used as a denominator.

Rule out two possibilities: 0 and negative number, that is, for x>0, then A can be any real number;

The possibility of 0 is ruled out, that is, for X.

The possibility of being negative is ruled out, that is, for all real numbers with x greater than or equal to 0, a cannot be negative.

To sum up, we can draw that when a is different, the different situations of the power function domain are as follows:

If a is any real number, the domain of the function is all real numbers greater than 0;

If a is a negative number, then X must not be 0, but the definition domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0.

When x is greater than 0, the range of the function is always a real number greater than 0.

When x is less than 0, only when q is odd and the range of the function is non-zero real number.

Only when a is a positive number will 0 enter the value range of the function.

Since x is greater than 0, it is meaningful for any value of a,

It must be pointed out that when x

Therefore, the power function in the first quadrant is given below.

You can see:

(1) All graphs pass (1, 1).

(2) When a is greater than 0, the power function monotonically increases, while when a is less than 0, the power function monotonically decreases.

(3) When a is greater than 1, the power function graph is concave; When a is less than 1 and greater than 0, the power function graph is convex.

(4) When a is less than 0, the smaller A is, the greater the inclination of the graph is.

(5)a is greater than 0, and the function passes (0,0); A is less than 0, and the function has only (0,0) points.

(6) Obviously, the power function has no boundary.