"Logarithmic Function" Exponential Function and Logarithmic Function PPT Courseware (Application of Logarithmic Function and Its Properties in Lesson 2)

Description

"Logarithmic Function" Exponential Function and Logarithmic Function PPT Courseware (Application of Logarithmic Function and Its Properties in Lesson 2)

Part One: Learning Objectives

1. Master the monotonicity of logarithmic functions, and be able to compare logarithms with the same base and logarithms with different bases. (emphasis)

2. Through the study of exponential functions and logarithmic functions, deepen your understanding of the meaning and role of two important mathematical ideas: classification discussion and combination of numbers and shapes. (emphasis)

core competencies

1. Cultivate logical reasoning literacy by learning the application of monotonicity of logarithmic functions.

2. Improve logical reasoning and mathematical operation literacy by learning the comprehensive application of the properties of logarithmic functions.

Logarithmic function PPT, part 2: cooperative exploration to improve literacy

Compare logarithmic values

[Example 1] Compare the following sets of values:

(1)log534 and log543;

(2)log132 and log152;

(3)log23 and log54.

[ Solution

Method 2 (intermediate value method): Because log534<0, log543>0, so log534

(2) Method 1 (monotonicity method): Since log132=1log213, log152=1log215,

And because the logarithmic function y=log2x is an increasing function on (0, +∞),

And 13>15, so 0>log213>log215,

So 1log213<1log215, so log132

Method 2 (Image method): As shown in the figure, draw the images of y=log13x and y=log15x in the same coordinate system. It is easy to see from the figure: log132

(3) Take the middle value 1,

Because log23>log22=1=log55>log54,

So log23>log54.

regular method

Common ways to compare logarithmic values

1Use the monotonicity of logarithmic functions with the same base.

2Use the graph of the logarithmic function or use the base-changing formula to convert the same real number.

3The base and the real number are different, find the middle quantity.

Reminder: When comparing the size of a number, first use properties to compare the size with zero or 1.

Solving Logarithmic Inequalities

[Example 2] It is known that the function f(x)=loga(x-1), g(x)=loga(6-2x) (a>0, and a≠1).

(1) Find the domain of function φ(x)=f(x)+g(x);

(2) Try to determine the value range of x in the inequality f(x)≤g(x).

[Ideas Enlightenment] (1) Directly use the set of simultaneous inequalities where the real number of the logarithm is greater than 0 to find the value set of x.

(2) Solve the inequality by dividing a>1 and 0

regular method

Three common types of logarithmic inequalities

1Inequalities of the form logax>logab can be solved with the help of the monotonicity of y=logax. If the value of a is uncertain, it needs to be discussed in two cases: a>1 and 0

2For inequalities of the form logax>b, b should be converted into a logarithmic form with a as the base, and then solved using the monotonicity of y=logax;

3Inequalities of the form logax>logbx can be solved using images.

Class summary

1. Comparing the size of two logarithmic values ​​and solving logarithmic inequality problems are based on the monotonicity of the logarithmic function. If the base of the logarithm is a letter and the range is unclear, it is generally divided into two categories: a>1 and 0

2. When solving problems related to logarithmic functions, we must establish the principle of "domain priority" and pay attention to the application of the idea of ​​combining numbers and shapes and the idea of ​​classification discussion in solving problems.

Logarithmic function PPT, the third part: reaching the standard in class and solidifying the double base

1. Thinking and analysis

(1) y=log2x2 is an increasing function on [0, +∞). ()

(2) y=log12x2 is an increasing function on (0, +∞). ()

(3) The solution set of ln x<1 is (-∞, e). ()

(4) The value range of function y=log12(x2+1) is [0, +∞). ()

2. Assume a=log32, b=log52, c=log23, then ()

A． a>c>b

B． b>c>a

C. c>b>a

D. c>a>b

3. The monotonically increasing interval of function f(x)=log2(1+2x) is ______.

4. It is known that a>0 and the inequality 22a+1>25a-2 is satisfied.

(1) Find the value range of the real number a;

(2) Find the solution set of the inequality loga(3x+1)

(3) If the function y=loga(2x-1) has a minimum value of -2 in the interval [1,3], find the value of the real number a.

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Update Time: 2024-09-29

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