The general form of logarithmic function is that it is actually the inverse of exponential function. Therefore, 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) obvious logarithmic function.

2. The first volume of senior three mathematics needs a knowledge point induction.

1, exponent, logarithm,

2.( 1) mapping is "holomorphism" plus "one arrow and one carving"; The elements in the first set in the mapping must have images, but the elements in the second set may not necessarily have original images (the image of the intermediate element is only the next one, but the original image of the intermediate element may not be available or may be arbitrary); The function is "mapping on a set of non-empty numbers", where "the range is a subset of the image set in the mapping",

(2) The function image has at most one common point with the vertical axis, but it may not have any common point with the vertical axis, or it may be any one.

(3) The function image must be a curve in the coordinate system, but the curve in the coordinate system is not necessarily a function image.

3. Monotonicity and parity

If (1) odd function is monotonic in an interval symmetrical about the origin, its monotonicity is exactly the same; if even function is monotonic in an interval symmetrical about the origin, its monotonicity is just the opposite.

(2) The monotonicity of the composite function is: "The same sex increases, and the same sex increases; When the opposite sex is reduced, the difference is different. The parity characteristic of composite function is "even inside is even, odd inside is the same as outside", and the change of definition domain should be considered in composite function. (that is, it makes sense to get back together)

4. Symmetry and periodicity (the following conclusions should be digested and absorbed, not memorized)

The (1) function and the image of the function are symmetrical about the straight line (axis).

Extension 1: If the function holds for everything, then the image is symmetrical about the line (determined by "half of the sum").

Extension 2: The image of the function is symmetrical about a straight line,

(2) Functions and function images are symmetrical about a straight line (axis),

(3) the function and the image of the function are symmetrical about the center of the coordinate origin,

3. The first volume of senior three mathematics needs a knowledge point induction.

(1) inevitable event: the event that will happen under condition S is called the inevitable event relative to condition S;

(2) Impossible events: events that will not happen under condition S are called impossible events relative to condition S;

(3) Deterministic events: inevitable events and impossible events are collectively referred to as deterministic events relative to condition S;

(4) Random events: events that may or may not occur under condition S are called random events relative to condition S;

(5) Frequency and number of times: repeat the test for n times under the same condition S, and observe whether there is an event A, and the number of times that the event A appears in the n tests is called event nA.

Frequency of occurrence of segment a; The ratio fn(A)=n when event a occurs.

Is the probability of occurrence of event A: for a given random event A, if the occurrence frequency fn(A) of event A is stable at a certain constant with the increase of test times, this constant is recorded as P(A), which is called the probability of event A. ..

(6) Difference and connection between frequency and probability: The frequency of a random event refers to the ratio of the number of times nA of the event to the total number of times n of experiments, which has certain stability and always swings around a certain constant, and the swing amplitude becomes smaller and smaller with the increase of the number of experiments. We call this constant the probability of random events, which quantitatively reflects the probability of random events. Frequency can be approximated as the probability of the event under the premise of a large number of repeated experiments.

4. The first volume of senior three mathematics needs a knowledge point induction.

The parity of 1. function (1) If f(x) is an even function, then f (x) = f (-x);

(2) If f(x) is odd function and 0 is in its domain, then f(0)=0 (which can be used to find parameters);

(3) The parity of the judgment function can be defined in equivalent form: f (x) f (-x) = 0 or (f (x) ≠ 0);

(4) If the analytic formula of a given function is complex, it should be simplified first, and then its parity should be judged;

(5) odd function has the same monotonicity in the symmetric monotone interval; Even functions have opposite monotonicity in symmetric monotone interval;

2. Some questions about compound function.

Solution of the domain of (1) composite function: If the domain is known as [a, b], the domain of the composite function f[g(x)] can be solved by the inequality a≤g(x)≤b; If the domain of f[g(x)] is known as [a, b], find the domain of f(x), which is equivalent to x∈[a, b], and find the domain of g(x) (that is, the domain of f(x)); When learning functions, we must pay attention to the principle of domain priority.

(2) The monotonicity of the composite function is determined by "the same increase but different decrease";

3. Function image (or symmetry of equation curve)

(1) Prove the symmetry of the function image, that is, prove that the symmetry point of any point on the image about the symmetry center (symmetry axis) is still on the image;

(2) Prove the symmetry of the image C 1 and C2, that is, prove that the symmetry point of any point on C 1 about the symmetry center (symmetry axis) is still on C2, and vice versa;

(3) curve C 1: f (x, y) = 0, and the equation of symmetry curve C2 about y=x+a(y=-x+a) is f(y-a, x+a)=0 (or f(-y+a,-x+a) =

(4) Curve C 1:f(x, y)=0 The C2 equation of the symmetrical curve about point (a, b) is: f(2a-x, 2b-y) = 0;

(5) If the function y=f(x) is constant to x∈R, and f(a+x)=f(a-x), then the image y=f(x) is symmetrical about the straight line x=a;

(6) The images of functions y=f(x-a) and y=f(b-x) are symmetrical about the straight line x=;

4. The periodicity of the function

(1)y=f(x) for x∈R, f(x+a)=f(x-a) or f (x-2a) = f (x) (a >: 0) is a constant, then y=f(x) is a period of 2a.

(2) If y=f(x) is an even function, and its image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 2 ~ a;

(3) If y=f(x) odd function, and its image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 4 ~ a;

(4) If y=f(x) is symmetric about points (a, 0) and (b, 0), then f(x) is a periodic function with a period of 2;

(5) If the image of y=f(x) is symmetrical (a ≠ b) about straight lines x = a and x = b, then the function y = f (x) is a periodic function with a period of 2;

(6) When y=f(x) equals x∈R, f(x+a)=-f(x) (or f(x+a)=, then y = f (x) is a periodic function with a period of 2;

5. Equation

(1) equation k=f(x) has a solution k∈D(D is the range of f(x));

(2)a≥f(x) considers A ≥ [f (x)] max;

A≤f(x) considers a ≤ [f (x)] min;

(3)(a & gt； 0，a≠ 1，b & gt0，n∈R+)；

logaN =(a & gt； 0，a≠ 1，b & gt0，b≠ 1)；

(4)logab symbols are memorized by the formula of "same positive but different negative";

a Logan = N(a & gt； 0，a≠ 1，N & gt0);

Step 6 draw pictures

When judging whether the corresponding relationship is a mapping, we should grasp two points:

The elements in (1)A must all have images and;

(2) All elements in B may not have original images, and different elements in A may have the same images in B;

7. Monotonicity of functions

(1) can skillfully use definitions to prove monotonicity of functions, find inverse functions and judge parity of functions;

(2) According to monotonicity, the problem of finding the range of a class of parameters can be solved by using the sign-preserving property of linear functions on intervals.

5. The first volume of senior three mathematics needs a knowledge point induction.

First, the problem of finding the domain of a function ignores that the domain of a detailed function is the range of independent variables that make the function meaningful. If candidates want to find the definition domain accurately in the examination room, they should find out the restrictive conditions of independent variables in various situations according to the resolution function and list them into inequality groups. The solution set of inequality groups is the definition domain of the function. When finding the domain of a general function, we should pay attention to the following points: the denominator is not 0; Even times are nonnegative open; The real number is greater than 0, and the power of 0 is meaningless. The domain of a function is a set of non-empty numbers, so don't forget this when solving the problem of the domain of a function. For compound functions, it should be noted that the domain of external function is determined by the domain of internal function.

Second, it is wrong to judge the monotonicity of a function by its absolute value. Functions with absolute values are essentially piecewise functions. There are two ways to judge the monotonicity of piecewise function:

Firstly, according to the monotonicity of the function expressed by the analytical formula of the function, the monotone interval is obtained on each segment, and then the monotone interval on each segment is integrated;

Second, draw the image of this piecewise function, and make an intuitive judgment by combining the image and nature of the function. Function problems cannot be separated from function images, which reflect all the properties of functions. When solving function problems, candidates should draw function images in their minds at the first time, analyze and solve problems from the images.

For monotone increase (decrease) intervals of different functions, remember not to use union, just indicate that these intervals are monotone increase (decrease) intervals of functions.

Third, the common mistakes in finding function parity The most common mistakes in finding function parity are: finding the wrong function definition domain or ignoring the function definition domain, unclear preconditions for function parity, improper judgment methods for piecewise function parity, and so on. To judge the parity of a function, we must first consider the domain of the function. The necessary condition for a function to have parity is that the domain interval of the function is symmetric about the origin. If this condition is not met, the function must be an odd or even function. On the premise that the domain interval is symmetrical about the origin, the judgment is made according to the definition of parity function.

When judging by definition, we should pay attention to the arbitrariness of independent variables in the definition domain.

Fourth, the reasoning of abstract functions is not rigorous. Many abstract function problems are designed by abstracting the * * * and "characteristics" of a certain function. When solving this kind of problems, candidates can solve abstract functions by analogy with the properties of some specific functions in this kind of functions. Using special assignment method, we can find the invariant property of function through special assignment, which is often the breakthrough of the problem.

The proof of the properties of abstract functions belongs to algebraic reasoning. Like the proof of geometric reasoning, candidates should pay attention to the rigor of reasoning when answering questions. Every step should have sufficient conditions, don't leave out conditions, and don't make up conditions. The reasoning process is distinct, and attention should be paid to writing norms.

Fifthly, the function zero theorem is not applied properly. If the image of the function y=f(x) in the interval [a, b] is the continuous curve of f(a)f(b) >

Sixth, the tangent of a point on the two tangent curves refers to the tangent of the curve with this point as the tangent point, and there is only one such tangent; Tangents of a curve passing through a point refer to all tangents of the curve passing through that point. If this point on the curve certainly includes the tangent of the curve at this point, there may be more than one tangent of the curve passing through this point.

Therefore, when solving the tangent problem of curve, candidates should first distinguish what is tangent.

Seventh, confuse the relationship between derivative and monotonicity. Function is a question type that increases function in a certain interval. If the examinee thinks that the derivative function of the function is always greater than 0 in this interval, it is easy to make mistakes.

When solving the relationship between monotonicity of a function and its derivative function, we must pay attention to the necessary and sufficient condition that the derivative function of the function monotonically increases (decreases) in a certain interval is that the derivative function of the function is constant (small) or equal to 0 in this interval, and the derivative function is not constant in any subinterval of this interval.

Eighth, the relationship between derivative and extreme value is unclear. When solving the function extreme value problem with derivative, candidates are prone to make mistakes, that is, they find the points that make the derivative function equal to 0, but they don't judge the signs of the derivative functions on the left and right sides of these points. They mistakenly think that the point where the derivative function is equal to 0 is the extreme point of the function, and often make mistakes. The reason for the error is that the examinee is not clear about the relationship between derivative and extreme value. The zero derivative function value of a differentiable function at a certain point is only a necessary condition for the function to take the extreme value at that point. I would like to remind the majority of candidates that when calculating the extreme value of a function with derivatives, we must carefully check the extreme point.