Logarithmic Equation: ( Log AX : Condition: 0  A  1 And X &gt; 0 ) Method 1: Same Base: Log AX  B  Log AYX = Y

Note: * Set t   a  b  x  0   a  b  x  1

* There are 3 groups containing x => divided by the group with the largest or smallest base

4. Logarithmic function: y = log a x

+ TXD: x > 0

+ The function is uniform when a > 1

+ Inverse function when 0 < a < 1

5. PT Mǜ

Method 1: Same base: a x =b = a y <=> x = y

Method 2: Not with the same base: a x =b <=> x = log a b

Method 3: Set the secondary hidden: Aa 2x +ba x +C = 0

+ Set t  a x ; t > 0

+ Solve for t => x

best

6. Logarithmic equation: ( log a x : Condition: 0  a  1 and x > 0 ) Method 1: Same base: log a x  b  log a y <=> x = y

Method 2: Not with the same base: log a x  b <=> x = a b

Method 3: Set the secondary hidden: A. log 2 x  B .log x  C  0

+ Set t  log a x ; there is no condition of t

+ Solve for t => x

7. Real pt mǜ and logarithm

Note: + Base a > 1: no direction change

+ Base 0< a < 1: change direction

8. Compound interest: A  a (1  r ) n

3. Teach new lessons

Teacher's Activities

Student Activities	Content
Activity 1. Investigate the properties of exponential functions, mǜ and mǜ functions logarithmic function
H1. Classify functions and state the conditions for their determination?	D1. a) 3 x  3  0  D = R {1} b) x  1  0 2 x  3  D = (  ;1)   3 ;      2  c) x 2  x  12  0  D = (  ;  3)  (4;  ) d) 25 x  5 x  0  D = [0; +∞)	1. Find the domain of the function a) y  1 3 x  3 b) y  log x  1 2 x  3 c) y  log x 2  x  12 d) y  25 x  5 x
Activity 2. Consolidate logarithm calculations
H1. What rules should be used?	D1. a) log a x = 8 b) log a x = 11	2. Give log a b  3, log a c   2 . Calculate log a x with: a) x = a 3 b 2 c b) x = a 4 3 b c 3
Activity 3. Solve equations, mǜ inequalities, logarithms
H1. If the solution?	D1. a) Return to base 3 and 5.  3  x  5  3       x = –3  5   3 	4. Solve the equations
		following submission:
		a)
		3 x  4  3.5 x  3  5 x  4  3 x  3

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Teacher's Activities

Student Activities

Content

 Note: x > 1

 log 7 x  0 .

H2. How to solve?

b) Divide both sides by 16 x .

 3  x

Let t    , t > 0.

 4 

 x = 1

c) log 7 ( x  1)  0  x = 8

d) log 3 x  3  x = 27

D2.

a) Return to the same base 2 .

 2  x

Let t    , t > 0.

 5 

2 t 2  3 t  5  0 

t  5

 x < –1.

b) Set t  log 0.2 x .

t 2  5 t  6  0  2 < t < 3

 0.008 < x < 0.04.

b) 4.9 x  12 x  3.16 x  0

log 7 ( x  1) log 7 x  log 7 x

log 3 x  log 3 x  log 1 x  6

5. Solve the following inequalities:

a) (0, 4) x  (2.5) x  1  1.5

log 2 x  5log x   6

0.2 0.2

Activity 4. Consolidation

Emphasize:

- Calculate and change formulas

- How to solve equations, inequalities and logarithms.

- Apply to general problems

IV. LEARN EXPERIENCE, SUPPLEMENT

+ Students take a long time to solve equations and often make mistakes

+ Students cannot solve the inequality.

APPENDIX 4

TEACHER: REVIEW CHAPTER II (EXPERIMENTAL CLASS)

I. OBJECTIVES

Knowledge

 Exponential, mǜ and lgarithmic functions.

 Rules for calculating factorials, mǜ and logarithms.

 Solve equations, inequalities and logarithms

Skills

 Investigate exponential functions, mǜ functions, and logarithmic functions.

 Calculate expressions: Factor, mǜ, logarithm

 Solve equations and inequalities.

 Practice solving math problems using Casio FX 570 calculator

Attitude

 Train carefulness, accuracy, quick reflexes, and quick prediction.

Think mathematical problems logically and systematically.

II. PREPARE

Teacher: Lesson plan. Exercise system.

Students: Textbooks, notebooks. Review all knowledge of chapter II

III. TEACHING ACTIVITIES

1. Stabilize the organization : Check class size.

2. Check the lesson:

1. Basic formulas

 a  n  1

a n

 a lo g a b  b ;

 log a  

 



 a  . a   a   

 log b    log b

 log a c  log a b  log a c

 m a n  a m

 a  

  

a 

 ( ab )   a  . b 

 log  b  log a b

a 

first

 log ( b . c )  log b  log c

  b 

 a  

a 

 



b 

 ( a m ) n  a m . n  ( a n ) m

Change the base

 log b 

first

log a

 log b  log c b

log a

Derivative

 ( x n )' = nx n-1

 ( e x )' = e x

 ( a x )' = a x .lna

 ( u n )' = nu n-1 .u'

 ( e u )' = e u .u'

 ( a u )' = a u .lna.u'

 ( log x )' =

first

x .ln a

 ( log u )' =

1 . u '

u.ln a

2. Definite set of redundant functions: y = u 

+  positive integer (Z + ) => u belongs to R

+  vowel or equal to 0 => u is different from 0

+  is not integer => u > 0

3. Function mǜ : y = a x

+ TXĐ: D = R

+ The function is uniform when a > 1

+ Inverse function when 0 < a < 1

; y = log a x

4. Logarithmic function: y = log a x

+ TXD: x > 0

+ The function is uniform when a > 1

+ Inverse function when 0 < a < 1

5. PT, BPT mǜ and logarithm

Method 1: Substitute the input with the CALC command

Method 2: Use the computer's search table with the command MODE 7 Note: + Base a > 1: does not change direction

+ Base 0< a < 1: change direction

6. Compound interest: A  a (1  r ) n

3. Teach new lessons

Activities of teachers

Student Activities

Content

Activity 1. Investigate the properties of exponential functions, mǜ and mǜ functions

logarithmic function

Ask. Show how to choose the formula

HD 1:

Method 1: Students remember

Method 2 : Calculator Choose x =2; m =3; n =4;

y =5

Pressing: 2 3 .2 4  2 3  4 will result in 0. So A is correct. Same for answers B and C

HD 2 : Similar

Select recipe

1. Which of the following equations is incorrect?

A. x m . x n  x m  n

B.  x n  m  x n . m

C.  xy  n  x n . y n

D . x m . y n   xy  m  n .

2. Find the correct statement in the following statements:

A. log a  x  y   log a x  log a y

Activities of teachers

Student Activities

Content

HD 3 : according to the formula

HD 4 : How to find it?

Students answer: x > 0

HD 5:

Method 1: Calculate Method 2: Calculator Choose x =2;

a 2  x  a  2

b  3 2 ; c  4 2

Click on the expression to find answer D

HD 6 : similar to HD 5

C. log 1  1

a x log x

3. The function y  log x has a definite set

A. D  R B. D  R  0 

C . D   0;   D. D    ;0 

4. The function y  x  2 has a domain as:

A. D    ;0  B. D  R

C. D   0 ;   D . D  R  0 

5. Given log a x  2, log b x  3, log c x  4 .

Calculate the value of the expression: log x 5

a 2 b c

A. 12 B. 1 C. 6 D. 24

13 9 13 7

6 .The value of the expression

a b  2  a b  1   a  3 b 3 

x  2 y  x  2 y  1 

To be:

A. _ a 8 b 2 B . a 2 b 8 C . a 8 D . a 8 . b

Activity 2. Train students to use computers

Logarithmic Equation: ( Log AX : Condition: 0  A  1 And X > 0 ) Method 1: Same Base: Log AX  B  Log AYX = Y

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